DIFFERENTIAL AND INTEGRAL CALCULUS WITH EXAMPLES AND APPLICATIONS pdf - Web Education

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(5) DIFFERENTIAL AND INTEGRAL. CALCULUS WITH EXAMPLES AND APPLICATIONS. BY. GEORGE. OSBORNE,. A.. WALKER PROFESSOR OF MATHEMATICS. IN. S.B.. THE MASSACHUSETTS. INSTITUTE OF TECHNOLOGY. REVISED EDITION. BOSTON, D.. C.. HEATH &. U.S.A.. CO.,. 1906. PUBLISHERS.

(6) ao2>. a. A. UBRARY of CONGRESS Two. Copies Received. JAN 111907. A. Cooyriarht Entry. Kioto. L.. It).. iSS. o^ XXc„. COPY. No.. B.. Copyright,. 1891. By GEORGE. A.. and. 1906,. OSBORNE.

(7) PREFACE In the original work,. tile. author endeavored to prepare a text-. book on the Calculus, based on the method of. limits, that. should. be within the capacity of students of average mathematical ability. and yet contain. all. that. is. essential to a. working knowledge of the. subject.. In the revision of the book the same object has been kept in view.. Most. the. of. text. been rewritten, the. has. demonstrations have. been carefully revised, and, for the most part, new examples have been substituted for the of subjects in a. There has been some rearrangement. old.. more natural. order.. In the Differential Calculus, illustrations of the " derivative" aave been introduced in Chapter "ion will. be found,. also,. II.,. and applications of. among the examples. differentia-. in the chapter. imme-. diately following.. Chapter VII.. on Series,. is. entirely new.. In the Integral Calculus,. immediately after the integration of standard forms, Chapter XXI. has been added, containing simple applications of integration.. In both the Differential and Integral Calculus, examples. illustrat-. ing applications to Mechanics and Physics will be found, especially in Chapter X. of the Differential Calculus, on. and. in. ter has. Maxima and Minima,. Chapter XXXII. of the Integral Calculus. been prepared by. my. The. latter chap-. colleague, Assistant Professor N.. It.. George, Jr.. The author. also acknowledges his special obligation to his col-. leagues, Professor H.. W. Tyler and. important suggestions and criticisms.. Professor F.. S.. Woods,. for.

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(9) CONTENTS DIFFERENTIAL CALCULUS CHAPTER. I. Functions PAGES. AF.T*.. I.. -7, 9. 8.. Variables and Constants Definition. and. 1. Classification of. Examples. Notation of Functions.. CHAPTER Limit. 10.. Definition of Limit. 11.. Notation of Limit. 12.. Special Limits (arcs. 13-15. 16.. 17-21. 22.. Increment,. 1-5. Functions. Increment.. 5-7. II. Derivative 8 8. and chords, the base. e). 8-10. .. Expression for Derivative. Derivative.. Illustration of Derivative.. .. .. .11,. Examples. Three Meanings of Derivative Continuous Functions. Discontinuous Functions.. CHAPTER. 12. 13-15 10-21. Examples. .. 22-25. III. Differentiation Algebraic Functions.. Examples. ..... Logarithmic and Exponential Functions.. Examples. Trigonometric Functions. Examples Inverse Trigonometric Functions. Examples .. Relations between Certain Derivatives.. Examples. 26-39. 39-45 45-51. 51-57. 57-60. CHAPTER IV Successive Differentiation 57, 58.. Definition and Notation. 60.. The nth. 60.. Leibnitz's Theorem.. Derivative.. Examples Examples. 61. 63-65 65-67.

(10) CONTENTS. VI. CHAPTER V Differentials.. Infinitesimals PAGES. 61-63.. Definitions of Differential. 64.. Formulae for Differentials.. 65.. Infinitesimals. 68-70. Examples. 71-73. .. 73,74. CHAPTER VI Implicit Functions 66.. Examples. Differentiation of Implicit Functions.. CHAPTER Series.. 75-77. VII. Power. Series. Convergent and Divergent Series. Positive and Negative Terms. Absolute and Conditional Convergence 69-71. Tests for Convergency. Examples Convergence of Power Series. Examples 72, 73. Power Series.. 67,68.. .. CHAPTER. .. 78, 79. .. 85-87. 79-85. VIII. Expansion of Functions 74-78.. Maclaurin's Theorem.. Examples. 88-93. Huyghens's Approximate Length of Arc 80,81. Computation by Series, by Logarithms 82. Computation of -w 83-87. Taylor's Theorem. Examples. 93 94-96. 79.. 96,97 97-100. .. 89.. 90-93. 94.. Rolle's. Theorem. Remainder. 101. .. Mean Value Theorem. ..... 101-104. .. 105. CHAPTER IX Indeterminate Forms 95.. 96, 97.. 98-100.. Value. 106. of Fraction as Limit. Evaluation of. 106-110. Examples. Evaluation of g, 0- oo, oo- ex),. 0°,. 1",. oo°.. Examples. 110-113.

(11) CONTESTS. vn. CHAPTER X Maxima and Minima of Functions of One Independent Variable PAGES. AUTS.. 101.. 102-104.. Definition of. Maximum and Minimum. Conditions for. Maxima and Minima,. 105.. When. 100.. Maxima and Minima by. Values. Examples. .. .. .. .. .. .. 114 114-119. Examples. 119-121. d.r. Taylor's Theorem.. Problems. 121-129. CHAPTER XI Partial Differentiation. Two. More Independent Variables Examples. ..... 107.. Functions of. 108.. Partial Differentiation.. 109.. Geometrical Illustration. 110.. Equation of Tangent Planes.. or. 130. 131. 131. 132. 133. Angle with Coordinate Planes. Examples Ill, 112.. 114-116.. Order of Differentia. Examples. tion.. 113.. 133-136. Partial Derivatives of Higher Orders.. Total Derivative. Differentiation. 136-139. Examples. Total Differential.. of Implicit Functions.. 140-144. Taylor's Theorem. Examples. 144-147. CHAPTER. XII. Change of the Variables 117.. 118. 119.. 120,121.. Change Independent Variable x Change Dependent Variable Change Independent Variable z. in. Derivatives 148, 149. to y. 149. Examples Derivatives from Rectangular. Transformation of Partial Polar Coordinates. to. z.. .. .. 150-152. to. 152-154. CHAPTER. XIII. Maxima and Minima of Functions of Two or More Variables 122,123. 124.. Definition.. Conditions for. Maxima and Minima. Functions of Three Independent Variables. 155, 156. 156-161. .. CHAPTER XIV Curves for Reference 120-127.. Cirsoid.. "Witch.. Folium. of Descartes. .. 162, 163.

(12) CONTENTS. Vlll. PAGES. ARTS.. Cubical Pa-. Parabola referred to Tangents. Semicubical Parabola. 128-130.. Catenary.. 131-134.. Epicycloid.. 135-145.. Spiral of Archimedes. HyperCircle. Polar Coordinates. Logarithmic Spiral. Parabola. Cardioid. bolic Spiral.. rabola.. Hypocycloid.. (-)*+{?)* =1. Equilateral Hyperbola. r. Lemniscate.. 164, 165 a*2/. -. 2. =« 2^-«6. 166,167. Four-leaved Rose.. = asm*-. 167-172. o. CHAPTER XV Direction or Curves.. Tangents and Normals. Subtangent. Subnormal. Intercepts of Tangent Angle of Intersection of Two Curves. Examples 148. Equations of Tangent and Normal. Examples 149-151. Asymptotes. Examples Polar Coordinates. Polar Subtangent 152, 153. Direction of Curve. and Subnormal 154. Angle of Intersection, Polar Coordinates. Examples 155, 156. Derivative of an Arc 146.. .. .. 147.. .. .. .. .. .. .. .. 173 174-176. 176-179. 179-182 182, 183. 183-186 186-188. CHAPTER XVI Direction op Curvature. 157. 158.. Points of Inflexion. Concave Upwards or Downwards Point of Inflexion. Examples. 189. 190-192. CHAPTER XVII Curvature.. Radius of Curvature. Involute. 157-161.. Curvature, Uniform, Variable. 162-164.. Circle of Curvature.. 193, 194. Radius of Curvature, Rectangular Co-. ordinates, Polar Coordinates. 165.. Evolute and Involute. 168-170.. Properties of Involute aud Evolute.. 173.. Order of Contact Osculating Curves. .. .. .. 195-200 200 201, 202. CHAPTER Order of Contact.. ...... Examples. Coordinates of Centre of Curvature. 166, 167.. 171,172.. Evolute and. Examples. .. .. .. 202-205. XVIII. Osculating Circle 206-208 208,209.

(13) CONTENTS. IX PAGES. Order of Contact at Exceptional Points 175. To find the Coordinate of Centre, and Radius, of the Osculating Circle at Any Point of the Curve 174.. 209. .. ..... 170.. Osculating Circle at. Maximum. Minimum. or. 209-211. Ex-. Points.. amples. 211-213. CHAPTER XIX Envelopes 177.. Series of Curves. 214. Envelope. 178, 179.. Definition of Envelope.. 180-182.. Equation of Envelope Evolute of a Curve is the Envelope of. 183,. is. Tangent. 214,215 215-217. .. .. Examples. Normals.. its. 217-221. INTEGRAL CALCULUS CHAPTER XX Integration of Standard Forms 184, 185.. Definition of Integration.. 186-190.. Fundamental Integrals.. Elementary Principles. .. 223-225 225-240. .. Examples. Derivation of Formulae.. CHAPTER XXI Simple Applications of Integration. Integration 191,192. 195.. Area Examples. Derivative of Area. Illustrations.. of Curve.. Constant of. Examples. 241-244. .. 244-248. CHAPTER XXII Integration of Rational Fractions 194, 195.. Formulae for Integration of Rational Functions Operations. 196.. Partial Fractions. 197.. Examples Case II. Examples Case III. Examples Case IV. Examples. 198. 199.. 200.. Case. p reliminary 249 250 250-253. .. I.. 254-256 256-259 ». .. 260-262.

(14) CONTENTS CHAPTER. XXIII. Integration of Irrational Functions PAGES. 202.. Integration by Rationalization. 263. p. 203, 204.. Integrals containing (ax. 206, 207.. Integrals containing. 208.. +. &)«,. +. (ax. V± x + q z + 2. Examples Examples. b) s. b.. 263-266. .. 266-268. Integrable Cases. CHAPTER XXIV Trigonometric Forms readily Integrable 209-211.. Trigonometric Function and. 212, 213.. Integration of tan n x dx, cot" x dx, sec n xdx, cosec n xdx. its. Examples. Differential.. .. .. 270-272 273, 271. 214.. Integration of tanm x sec n x dx, cot m x co&ec n xdx.. 274-276. 215.. Integration of. 276-278. Examples sin™ x cos w x dx by Multiple Angles. Examples. CHAPTER XXV Integration by Parts.. by. 216.. Integration. 217.. Integration of. 218-222.. Parts. e ax sin. 279-282 dx.. Examples. .. .. Reduction Formulae for Binomial Algebraic Integrals. vation of Formulae.. 223, 224.. Reduction Formulae. Examples nx dx, e ax cos nx. 283,284. Deri-. Examples. 284-291. Examples. Trigonometric Reduction Formulae.. .. .. .. 291-294. CHAPTER XXVI Integration by Substitution p. 226.. Integrals of f(x 2 )xdx, containing (a. 227.. Integrals containing Vcfi Substitution.. 228-232.. y/x 2. Examples. bx2 )i.. ± a2. 295, 296. by Trigonometric. Examples. Integration of Trigonometric tion.. 233.. -f. 296-299. Forms by Algebraic. Substitu-. Examples. 299-304. Miscellaneous Substitutions.. Examples. 304, 305. .. CHAPTER XXVII Integration as a Summation. 234.. 235-237.. Integral the Limit of a. Area. of Curve.. Integral. Definite Integrals. Sum. Definite Integral.. 306. Evolution of Definite ,. 306-309.

(15) CONTEXTS. XI PAGES. ART?.. of Definite Integral.. 288, 239.. "Definition. 240-242.. Sign of Definite Integral.. 243-245.. Change. Constant of Integration.. Examples. 310-314 Infinite Limits.. Infinite. Values 314-317. of/(.r). Definite Integral as a. of Limits.. Sum. .. .. .. 317-319. CHAPTER XXVIII Application- of Integration to Plane Curves.. Application to Certain Volumes Areas of Curves, Rectangular Coordinates. Examples Areas of Curves. Polar Coordinates. Examples 249. Lengths of Curves. Rectangular Coordinates. Examples 250. Lengths of Curves, Polar Coordinates. Examples. 246. 247.. 24S.. 251. 252, 253.. .. Volumes. of Revolution.. Volumes by Area. .. .. .. .. Examples. Examples Examples. of Section.. 320-324 325-327 327-330 330-332. 333-335. Derivative of Area of Surface of Revolution. faces of Revolution.. 254.. .. .. Areas of Sur-. ..... 336-339 340-342. CHAPTER XXIX Successive Integration 255-25'. Definite Double Integral.. Examples. Variable Limits.. Triple Integrals.. 343-345. .. CHAPTER XXX Applications of Double Integration 258-262.. Moment. Double Integration, Rectangular CoPlane Area as a Double Examples. of Inertia.. ordinates. Integral.. Variable Limits.. Double Integration, Polar Coordinates. Moment of Inertia Variable Limits. Examples 266. Volumes and Surfaces of Revolution, Polar Coordinates Examples. 346-350. 26:5-265.. 350-353 353, 354. CHAPTER XXXI Surface, Volume, and 267.. To. Moment of Inertia of Ant Solid. Area of Any Surface, whose Equation between Three Rectangular Coordinates, x, y, find the. amples. is. given. z.. Ex355-360.

(16) CONTENTS. Xll. To. 268.. find the. Volume. whose Equation ordinates,. Moment. 269.. jc,. is. y, z.. of Inertia of. of. Any. Solid. bounded by a Surface,. given between Three Rectangular Co-. Examples. Any. 361-363. Examples. Solid.. .. .. .. 363, 364. CHAPTER XXXII Centre of Gravity. Pressure of Eldids. Eorce of Attraction Examples Examples Pressure of Liquids. Examples Centre of Pressure. Examples Attraction at a Point. Examples. 365-369. Centre of Gravity.. 270,271.. Theorems. 272, 273. 274.. 275. 276.. of Pappus.. .. 369, 370. .. 370-373 373-375 375-377. CHAPTER XXXIII Integrals for Reference. 277.. Index. .. .. ..... 378-385 386-388.

(17) DIFFERENTIAL CALCULUS CHAPTER. I. FUNCTIONS 1.. Variables and Constants.. unlimited number of values. is. A. quantity which. may assume an. called a variable.. A. quantity whose value is unchanged is called a For example, in the equation of the circle. x 2 +y°. =a. constant.. 2 ,. For as the point whose is a constant. moves along the curve, the values of x and y. x and y are variables, but a coordinates are. x, y,. are continually changing, while the value of the radius a remains. unchanged. Constants are usually denoted by the a, b, C, a,. (3,. first letters. of the alphabet,. y, etc.. Variables are usually denoted by the last letters of the alphabet, *, y, z,. 2.. <t>,. «A,. etc. -. Function.. When. one variable quantity so depends upon an-. other that the value of the latter determines that of the former, the. former. is. said to be a function of the latter.. For example, the area of a square is a function of its side the volume of a srjhere is a function of its radius the sine, cosine, and ;. ;. tangent are functions of the angle x2 are functions of. x.. ,. log. (V9. ;. the expressions. + 1), V*(* + l),.

(18) DIFFERENTIAL CALCULUS. 2. A quantity may be a function of two or more variables. For example, the area of a rectangle is a function of two adjacent sides; either side of a right triangle is a function of the two other sides the volume of a rectangular parallelopiped is a function of its three dimensions.. The expressions x 2 + xy. and. are functions of x. +y. g(x2 + y2), a x+ ^,. 2 ,. \. y.. The expressions. + yz + zx, ^^~~f. xy are functions of 3.. and. x, y,. log(x 2. + y~z),. z.. Dependent and Independent Variables.. If y. is. a function of. x,. as in the equations. y. x. =x. 2 ,. y. = tan. 4. y. a?,. =e. x -f-. and y the dependent variable. a function of x, x may be also regarded. called the independent variable,. is. It is evident that. as a function of. y,. variables reversed.. when y is. and the positions of dependent and independent Thus, from the preceding equations,. x=Vy,. x. = ±tan-. 1. y,. x. = log. e. (y-l).. In equations involving more than two variables, as z. + x — y = 0,. iv. + wz + zx + y = 0,. one must be regarded as the dependent variable, and the others as independent variables. Algebraic and Transcendental Functions. An algebraic function one that involves only a finite number of the operations of addi-. 4. is. tion, subtraction, multiplication, division, involution. with constant exponents.*. and evolution. All other functions are called transcen-. Included in this class are exponential, logarithmic,. dental functions.. and inverse trigonometric, functions. Note. The term "hyperbolic functions" is applied to certain forms of exponential functions. See page 00.. trigonometric or circular,. —. *. A more general. tion to the variable. definition of Algebraic Function is. is,. a function whose rela-. expressed by an algebraic equation..

(19) FUNCTIONS. A. Functions.. Rational. 5.. integral powers of. x,. is. polynomial involving only positive an. called. integral function of x\. as,. for. 2 + x - 4 .r + 3 x\. example,. A. rational fraction is a fraction. whose numerator and denominator. are integral functions of the variable. a,r3 x*. A. 3. example,. as, for. ;. + 2q;-l. + x*-2x'. rational function of x is an algebraic function involving no frac-. powers of x or of any function of x. of such a function is the sum of an integral function and a rational fraction as, for example, tional. The most general form. ;. 2. 6.. Explicit. ,r. rf-. 3x -2x 2. x-1+. and Implicit. x-. +l. When. Functions.. one quantity. pressed directly in terms of another, the former explicit. is. ex-. function of the latter.. For example, y y. is. an explicit function of x in the equations. = a? + 2x,. y. "When the relation between y and x. = Va + 1. 2. is. given by an equation con-. taining these quantities, but not solved with reference to to be. is. said to be an. an. y,. y. is. said. implicit function of x, as in the equations. axy. -f. bx 4- cy. Sometimes, as in the. +d=. first of. y. 3. -\-. log y. these equations,. = x. we can. solve the. y, and thus change the function from Thus we find from this equation,. equation with reference to implicit to explicit.. _bx±d m. ax 7.. Single-valued. and. Many-valued y. for every value of x, there. Expressing x. in. +. terms of. = X- - 2. Functions.. In. y,. the. .r,. one and only one value of. is. x. c. we have. = 1 ± Vy + 1-. y.. equation.

(20) DIFFERENTIAL CALCULUS. 4. Here each value case,. y. is. In the. An. of y determines two values of a single-valued function of x.. x. latter case,. is. a two-valued function of. w-valued function of a variable x. values corresponding to each value of. The. 8.. Notation. and the is. of. like, are. x.. x,. we may. has an unlimited num-. x.. Functions. The symbols used to denote functions of. a function of x,". ?/.. a function that has n. is. inverse trigonometric function, tan -1. ber of values for each value of. In the former. x.. F(x),f(x), <£(#), \f/(x), Thus instead of " y. x.. write. y=f(x), or y. = <f>(x).. A functional symbol occurring more than once in the same problem or discussion is understood to denote the same function or Thus. operation, although applied to different quantities. •. f(x). then. f(y)=y*. if. = x* + 5,. + &,. /( ft ). (1). = a + 5, 2. + l) = (a + l) + 5 = a* + 2a + 6, /(1) = 6. /(2) = 2 + 5 = 9, 2. /(a. 2. In. all. these expressions. fined. by. (1). ;. that. /(. denotes the same operation as de-. ). the operation of squaring the quantity and. is,. adding 5 to the result. Functions of two or more variables are expressed by commas be-. tween the Thus if then. variables.. / (x,. 2. then. xy. - f,. = a + 3 ab - b\ f(b, a) = b + 3 ba - a jf(3, 2) = 3 + 3-3-2 - 2 = 23. f(a, 0) = a 4>(x, y, z) = x + yz-y + 2, l,-l) = 3 + l(-l)-l + 2 = 27; <f>(3, *(a, 0, 0) = a - b + 2 <K0, 0, 0) = 2.. f(a, b). 2. 2. 2. 2. 2. If. =x +3. y). .. 2. s. 2. 3. 3. 2. ;. 2 ..

(21) FUNCTIONS 9.. y. and. if. by. If y is a given function of x, represented. Inverse Function.. from. 5. this relation. +(*),. we express x '. then each of the functions. =. (i). <f>. =. and. in terms of. y,. so that (2). •/<(.'/),. \p. said to be the inverse of the. is. other. .. For example,. = * = <£<»> x =VS = ^O/)V. if. then. Here. \p,. the cube root function,. the. is. inverse. of. cf>,. the cube. function.. = a* = <Kx), x = log y = xp(y). V. If. then. a. Here. the logarithmic function,. «/r,. function.. x. y =-,. Again, suppose. is. the inverse of. <f>,. the exponential. n -4- 2i. — x-=<f>(x) v — 2 x = j = \p(y). (3). r. From. this. we derive. '. (4). Here xp as defined by (4) is the inverse of as defined by (3). The notation ^> _1 is often employed for the inverse function of <f>. Thus,. if. y. if. =. = x=f~ x. <£(#),. y=f(?),. The student. is. <£. 0/). 1. (y).. already familiar with this notation for the inverse. trigonometric functions. If. 1.. y. Given. <j>.. _1. =. x. sin x,. = sin -1. EXAM PLES 2x* — 2 xy + y = a 2. ?/.. 2 ;. change y from an implicit to an explicit function. y. = x ± Va — x 2. 2 ..

(22) DIFFERENTIAL CALCULUS. 6. Given. 2.. sin. (a;. ft. — y)~ m. sin. ?/. change y from an implicit to an explicit function, Ans.. ^3.. Given. that. +. =/(*). ft). / (a +. 1). + (6. -6. 2 a;. F(x). s. fc. .. ;. Show. / (2a?) -/(- 2 ag =. }. <£. Show. ,. 2. 2. = ^t^l:. If. 2. a;. 2. Griven.f(x) that. /(0).. 1),. =6 + l)h + (6x- 3)/i + 2. -/(*) i». ;. s. show that. %K^.. /(-. = (x -1) i^(a? + 1) - F(x — 1) = 8 X. H. Given. h.. m + cos x. = 2^-3^+^+2. /(a?). Show. —. 1. /(l), /(2), /(|),. find. fix. = tan'. y. (0). =. c»,. <£. find/(0),. (a -\-b). /(*)+/(-«).. [/ (a:)]. =. .. - [/ (- a?)]. 2. 2 .. cj>(a)cj> (6).. that the same relation holds for the function. = cos 4- V"^ sin (a) ^(6). if/(a + b) = $(0). giving. (9,. if,. " If. 7. >«-sfey. -. show that the inverse function 8.. If. <f>(x). =. X. ax—. ,. is. If. find the inverse function of. /(«). show that /-1 (a;) 10.. Show. / (a? +. ft,. <j>.. b. =. c.. = log.(«+'V?=l), = a* + *y •. / (a^) = ax* + 2 toy + q/. If. same form.. c. Compare the two functions when. V9.. of the. 2 ;. find /(l, 2),. / (y, - a?).. that. y. +. ft). =/(«,. 2/.). + 2(aa? + by)h + 2(bx + cy). ft. +f(h,. ft)..

(23) FUNCTIONS \m-\-n 11.. Given. «£(m,n). \m. where m, <f>. 12.. Given. (m, w. »,. |w. are positive integers;. + 1) +. <£ (m. / fe. show that f{y +. + 1,. & ») z,z. show that. ») = <K m + 1? + !)• ?i. z. «,. ?/,. 2,. X,. y. y». *>. X. + x, x + y) = 2/. (a;,. y, 0)..

(24) CHAPTER LIMIT. 10.. When. Limit.. II. INCREMENT. DERIVATIVE. the successive values of a variable x approach. a, in such a way that the difference becomes and remains as small as we please, the value a is. nearer and nearer a fixed value. x. —a. called the limit of the variable. The student meaning. x.. supposed to be already somewhat familiar with the. is. which the following. of this term, of. may. illustrations. be. mentioned.. The limit of the value of the recurring decimal number of decimal places is indefinitely increased, is The limit of the sum of the series 1 + + + -J-H -J-. of terms. The. is. indefinitely increased,. limit of the fraction. x. The is. .3333. —. -J-. ...,. as the. ^. ,. number. as the. is 2.. ~a —a. ,. as x approaches a, is 3. circle is the limit of a regular polygon, as the. a2. number. .. of sides. indefinitely increased.. The. limit of the fraction. —. —. ,. as 6 approaches zero,. is 1,. provided. 6. 6. is. expressed in circular measure.. 11.. Notation. "Lim^". of. Limit.. "The. denotes. Special Limits.. 2. (2 x. 2. 2. There are two important limits required. in the following chapter. (a). Lim 0=o. ,. will be used. a, of.". —a == 2. xr — ax - hx + h ) =2x>. x2. Lim A=0 Some. x approaches. Lim z=a —. For example,. 12.. The following notation. limit, as. 6 being in circular measure..

(25) ;. LIMIT Let the angle. ADA' =. 2. From geometry, ABA' that. 2 a sin. is,. <. that. 2a0. is,. sin. <2 a tan > cos. and. sin. a be the radius of the arc. let. ^-. 2 aO,. Also from geometry,. 0.. v. (i). ACA' < ADA sin. fl. 0,. ^1). and. >e, (2) is. (2),. mediate in value between. As. approaches zero, cos. Hence. ;. cos. 0.. Hence by. ACA!.. < AC A'. u 1. inter-. and cos. 0.. approaches. 1.. Lini„ =0. sin 9. ... The student and. sin. 6,. will do well to compare the corresponding values of taken from the tables, for angles of 5°, 1°, and 10'.. Angle. sin 6. 5°. — = .0872605 36. .0871557. = .0174533. .0174524. =.0029089. .0029089. 1°. -5180 10'. ' 1080. (b). Lim 2=ae. M + —IV z )' ]. Before deriving this limit. the value of the. ex pression for increasing values of (1 (1. Thus,. z.. + 1) = 2.25 + 1) = 2.48832 2. 5. + to) = 2.59374 = 2.70481 (1.01) = 2.71692 (1.001) = 2.71815 (1.0001) = 2.71827 (1.00001) = 2.71&18 (1.000001) (1. us compute. let. •. 10. 100. 1000. 10000. 100000. 1000000. ..

(26) DIFFERENTIAL CALCULUS. 10. The required last. limit will be found to agree to five decimals with the. number, 2.71828.. By. the Binomial Theorem,. which may be written. iy = +. 1\ /. /. 1 i. zj. When. ^. + i + izl +v [2. |3. ,. z increases,. the fractions -, z. T. have. This quantity. is. approach. zero,. and we. e,. so that. = 1+7 + %+£ + £ + '—1. The value. etc.,. z. usually denoted by e. 2. ~w~~z/. |2. [3. of e can be easily calculated to. |4. any desired number of. decimals by computing the values of the successive terms of this series.. For seven decimal pi ices the calculation. is. as follows:. 1.. 2) 1.. 3). .5. 4). .166666667. 5). .041666667. 6). .008333333. 7). .001388889. 8). •000198413. 9). .000024802. 10). .000002756. 11). .000000276 .000000025. e= This quantity. e is. 2.7182818--.. the base of the Napierian logarithms.. * For a rigorous derivation of this limit, the student tensive treatises on the Differential Calculus.. is. referred to. more ex-.

(27) DERIVATIVE. 11. An. increment of a variable quantity is any addidenoted by the symbol A written before this quantity. Thus Ax denotes an increment of x, Ay, an increment of y. For example, if we have given Increments.. 13.. tion to its value,. and. is. y. = 10, then. and assume x of y. is. if. we. =A. increase the value of x by. increased from 100 to 141, that. In other words,. if. is,. by. 2,. the value. 41.. we assume the increment. of x to be. A& = 2, we. increment of y to be A?/ = 44. If an increment is negative, there is a decrease in value.. shall find the. For example, calling x = 10 as before, in y = x. Ax. if. y. and the same. initial. value of. us calculate the values of. A.'-.. We. Ax =. It. zero.. =x. 2 ,. = 10,. Ay corresponding. to different values of. then Ay. =. and -^-. =. Ax. 3.. 09.. 23.. 2.. 44.. 22.. 1.. 21.. 21.. 20.1. 0.1. 2.01. 0.01. 0.2001. 20.01. 0.001. 0.020001. 20.001. 20 h. h. The. equation,. thus find results as in the following table.. If. ments. = - 36.. x,. z let. ,. Ay. then. With the same. Derivative.. 14.. = - 2,. 2. +. 20+. 1C-. ft. third column gives the value of the ratio between the increof. ./. and of. y.. appears from the table that, as. Ay. also diminishes. Ax diminishes and approaches. and approaches. zero..

(28) DIFFERENTIAL CALCULUS. 12. The. ratio. —^. proaches 20 as. its limit.. This limit of. and. is. ^= dx. 20.. diminishes, but instead of approaching zero, ap-. —^ is. called the derivative of y with respect to x,. denoted by -^. **. In. this case,. It will be noticed that the value. tion y. =x. 2 ,. Without. and partly on the. -^ from y = x2 dx Increase x by Ax.. the derivative. 20 depends partly on the func-. initial value. restricting ourselves to. tain. when x = 10,. 10 assigned to. any one. initial value,. x.. we may. ob-. .. Then the new value. of y will be. = (x + Ax) Ay = y — y = (x + Ax) — x = 2xAx + (Ax) 2. y'. ;. therefore,. Dividing by Ax,. The. limit of this,. Ay = 2x+ Ax. 2 .. ^. when Ax approaches. —=2. Hence,. 2. 2. 1. zero, is 2x.. x.. dx. The. derivative of a function. may. then be defined as. the limiting. value of the ratio of the increment of the function to the increment of the variable, as the latter increment approaches zero.. It is to be noticed that -2 is not here defined as & fraction, but as. dx a single symbol denoting the limit of the fraction will find as he advances that. —. has. many. A ^. —. The student. of the properties of. ordinary fraction.. The. derivative. is. sometimes called the. differential coefficient.. 15. General Expression for Derivative.. In general,. y -/(») Increase x by Ax, and. we have. the. new. value of y }. y'=f(x-\-Ax).. let. an.

(29) :. .. DERIVATIVE Ay =. 13. - y =/(* + A ^) — /(*)i A?/ = /(* + A.r)-/(.r). y'. A#. A.y. A# The process of. (f.i'. Geometrical Illustration.. finding the. derivative. from y = x-, may be illustrated by a squared Let x be the length of the side OP, and y the area of the square on OP.. That is. y is the number of square units corresponding to the linear unit of x.. When area. the side. is. Ay. increased by PP', the. increased by the space between the. is. y. X. squares.. That. Ay_. is,. Ay=2xAx+(Ax)* =^=2x+Ax, Ax dy ax. From. 16.. T. A?/. •. p. the definition of the derivative. process for obtaining. Ax. X. 2x.. Ax. we have. P'. the following. it. (a) Increase x by Ax, and by substituting x -f- Ax for x, determine y + Ay, the new value of y. (b) Find Ay by subtracting the initial value of y from the new. value. (c). (d). limit. Divide by Ax, giving. — Ax. Determine the limit of _ is. •. _?,. as. Ax approaches. zero.. —. the derivative. dx. Apply. 1. .. y. this process to the following examples.. =2. 3. .r. — 6 x + 5.. Increasing x by Ax,. EXAMPLES. we have. y + Ay = 2(x + Axf-6(x + Ax) + 5; therefore, Ay = 2 (x + Ax) — 6 (x + Ax) + 5 — 2 x' + 6x—5 = (6 x - 6) Ax + 6 x(Ax) + 2(A x) 3. 2. 2. 3. .. This.

(30) DIFFERENTIAL CALCULUS. 14 Dividing by Ax,. -^ = 6x -6 + 6xAx + 2(Ax) v } 2. 2 .. Ax. * = Lim A ^/ = 6x Ax dx 2.. y. y. = x. +1. x + Ax + Ax + 1 x + Ax Ay = x + Ax + 1. + Ay =. -6.. 2. x. Ax. x z. +1. (x-f Ax-|-l)(x. + l). Ay = 1 Ax (x + Ax-fl)(x + l)'. ^. = _J: ^-Lira ^°Ax (x + 1) dx~. 2. 2/= Vx.. 3.. y. -\-. Ay. = Vx -f Ax,. A?/. = Vx + Ax — Vx,. Ay _ Ax The. limit. Vx. of. -f-. Ax — Ax. this. V#. _. takes the. rationalizing the numerator,. indeterminate form. Az=0. ^4. 4. 2/ y=x. -2x + 3x-4,. 4 aj -. 5.. Ax. 2. ?/=(x-a) 3. ,. But by J. we have. Ay _ Ax Ax Ax(Vx + Ax+Vx) dx. -.. 1. Vx+Ax-fVx. 2V. ^ = 4x -4x + dx 3. ^/ = dx. 3(x-a) 2. .. 3..

(31) DERIVATIVE 6.. y=(« + 2)(3-2«). 11. g = -4*-l.. J. %. mo;. =. —X. it. x. 15. >. \n-xf. dx. = 2.V-T. »*. +. ». mn. da. 9. "<*. cty. = .r + a. + *>*. 2. 10.. .-/. dx. V. ,=. *12.. ^/=V^T2. dx _. 14.. 15.. 16.. y. 17.. dy. dx. "We shall. _. dt~ that the derivative of the area of a is its. that the derivative of the. now. 'circle,. '. X. -Vcr—x2 1. 2$ with respect. circumference.. respect to its radius,. tive.. Zx% 2. dy _ dx. =±. Show. =. dx. 2. Show. +l. dy _ 1 dx 2Vz-r-2. = x\. to its radius,. + a). dt. 2/=V« -^, s. «. ~. (*-!)". 8. 13.. («j. is. volume of a sphere,. with,. the surface of the sphere.. give some illustrations of the. meaning of the deriva-.

(32) DIFFERENTIAL CALCULUS. 16. 17. Direction of a Plane Curve.. This. is. one of the simplest and. most useful interpretations of the derivative. Let P be a point in a curve determined by and PT the tangent at P. = y. Let OM=x,. its. equation y. = f(x),. MP. If. Ax. we. give x the increment. = MN,. y will have the. Draw PQ.. Now. will. Then. Ax diminish and Ay will also. if. approach approach. in-. = PQ.. crement Ay. zero,. point. the. zero,. move along. PQ. towards P, and. Q. curve. the. will. approach in direction PT as (1), we have. its. limit.. Taking the limit of each member of. TPR = Lim Ax. tan. That. the derivative. is,. —. ,. at. ^ =^ Ax dx. _. any point of a curve,. is. the trigono. dx metric tangent of the clination to. tangent. OX. line. in-. of the. that. at. point.. This. quantity. The. de-. is. noted by the term. slope.. slope of a straight. line is the tangent of its. inclination. of. to. the. axis. X The. slope. of. a plane. curve at any point. is. slope of its tangent. the at. that point.. Thus, -^, at any point of a curve, point.. is. the slope of the curve at that.

(33) — DERIVATIVE For example, consider the parabola x2. The. At. slope of the curve 1. y. •. 2p. = 0, the slope = 0, the direction being x = 2 p, the slope = 1, corresponding to an. At L, where Beyond. L. all. horizontal.. inclination. X. the slope increases towards. towards the limit. For. —=—. dx. = -x— 2. = ±py,. where x. 0,. of 45° to the axis of. is. is. 17. oo,. the inclination increasing. 90°.. points on the left of. x. Y",. is. negative, and hence the slope. negative, the corresponding inclinations to the axis of. X. being. negative.. Velocity in. Terms. over the distance. OP =. 18.. of a Variable s in. A. denoting Time.. t. the time. t,. s. body moves. being a function of. t;. it is. required to express the velocity at the point P.. Let As denote the. PP. distance. in the interval At. it. £. ir. would be equal. P and. -. ±-,. P, and. is. were uniform during this interval,. If the velocity to. As. For a variable velocity,. make. —. traversed. —. As is. the average or. more nearly equal. mean. velocity between. to the velocity at. P the. less. we. At.. That. is,. the velocity at. — -r.' P — Lim Afe0 -r. At dt. If v denote this velocity, v. Thus,. —. is. — = ds. •. the rate of increase of. s.. dt. Similarly,. -vr. and. -rr. are the rates of increase of x. and y respectively..

(34) DIFFERENTIAL CALCULUS. 18 19.. The. Acceleration.. rate of increase of the velocity v is called. acceleration.. If. we denote. this. by. a,. we have by a. —. the preceding article,. — dv dt. For example, suppose a body moves so that s. Then the. v. velocity,. = t\ = ds. _. 3?. dt'. dv. and the acceleration,. 6t.. dt'. Rates. 20.. For further two following problems. of Increase of Variables.. derivative, consider the. illustrations of the. Problem 1. A man walks across the street from A to B at a uniform rate of 5 feet per second.. A lamp. at. L. throws his. shadow upon the wall MN. AB is 36 feet, and. BL. 4. feet.. How. fast is. the shadow moving when. he. 16 feet from. is. When. 26 feet. ?. A?. When. 30 feet? Let P and Q be simultaneous positions of man and shadow.. Then. When That. is,. y _ BL. x~PB. 4 36. he walks from. —x. P. Let. AP = x, AQ = y.. 4x V. (1). 36 —a;. to P', the. shadow moves from Q. to Q'.. when Ax=PP', \y=QQ'.. Let At be the interval of time corresponding. Then we may. write. Ay =. Ax. to. Ax and. Ay.. At^ '. Ax At. (2).

(35) DERIVATIVE If. now we suppose. Ay and A.i* will and we have for the limits of the two. At to diminish indefinitely,. also diminish indefinitely,. members. 19. of ^2),. f?. dy. — dt dx. dx. rate of increase of. ?/. rate of increase of. x. Art. 18.. dt. velocity of. That. shadow. at. is,. velocity of. Finding the derivative of. (1),. any point. Hence,. shadow. at. any point. See Ex.. 144. -. = = = Problem. The. 2.. —. -. ;. when x. ;. 20 feet per second, when. of a ladder 20 feet long rests against a wall.. moved away from the wall. 12 feet. from the wall ? AVhen 16 feet from the wall ? Suppose PQ to be one position of the ladder.. Let. AP=. x,. AQ = y.. Then y. = 16 = 26 x = 30.. when x. 7.2 feet per second,. is. = V400 - x. 2 .. man). feet per second. the top moving,. is. the foot. of. X)-. rate of 2 feet per second.. How fast. (velocity. 2. (5 feet per second). foot of the ladder is. when. — x}. 1.8 feet per second,. The top. Art. 16.. x). 720 (oo. 8,. 144. Q=. (36 (36. dy dx. we have. 144 dy = dx (36 - xj velocity of. Q_. man. (3). at a. uniform.

(36) DIFFERENTIAL CALCULUS. 20. When That. Pto. the foot moves from. is, if. Ax. In the same. = PP', way. Ay. =. as in. P', the top. moves from Q. to. Q\. QQ'.. Problem. 1,. Ay Ay Ax. _ At ~ Ax At~. dy. And from. dy. this,. _dt. dx~ dx' dt. that. Q_. velocity of top at is,. velocity of foot. From. _. dy (3),. dy dx. —x. See Ex. 14, Art. 16.. <^~V400-;/. Hence, velocity of top at any point. Q. =. = (velocity of foot). V400-X. 2. 2x. V400-^. feet per second. 2. The negative sign is explained by noticing from the figure that y when x increases. Hence the rates of increase of x and y. decreases. have different signs.. When. x. —. 12,. velocity of top. =—. 11 feet per second.. When. x. = 16,. velocity of top. =—. 2| feet per second.. From. these problems. it. the increments of *y and x, '. of these variables.. appears that, while. —. dv dx. is. Aw -^. is. the ratio between. the ratio betiveen the rates of J increase.

(37) DERIVATIVE Increasing and. 21.. junction of x. is. Decreasing Functions.. if. If the derivative. of a. when x increases; and function decreases ivhen x increases.. positive, the junction increases. the derivative is negative, the. For. 21. —. the derivative. ,. which. if. the ratio between the rates of. is. dx increase of the variables (see conclusion of Art. 20),. is positive, it. follows that these rates must have the same sign. is,. when x But. increases,. -^ ax. if. must have. negative, the rates increases,. different signs. ;. that. is,. and increases when x decreases.. also evident geometrically. is. y increases. and decreases when x decreases.. when x. y decreases. This. is. ;. that. by regarding -^. as the slope of. a curve.. A to B,. As we pass from C,. y increases as x increases, but from. B to. y decreases as x in-. creases.. A. Between slope. —. B. and. positive. is. ;. the be-. tween B and C, negative. In the former case y is said to be an increasing function. ;. the. in. latter. case, a decreasing function.. For example, consider = x3 from which we find -^ = ^x rh. the function y. Since. — dx. is. ,. positive for all values of. x,. 3x*.. the function y. =x. 3. is. an. increasing function. If. we take y. 1 — we ,. find. —. 1. dx. x. Here we have a decreasing function with a negative derivative. Another illustration is Ex. 1, Art. 16, y. When When. x. is. x. is. =2. 3. ar. - 6 x + 5,. ^ =6 ax. (a?. - 1).. numerically less than 1, y is a decreasing function. numerically greater than 1, y is an increasing function..

(38) DIFFERENTIAL CALCULUS. 22 22.. A. Continuous Function.. y=f(x), is said to be when y = f (afc) is a definite AxQ approaches zero, Ax being. function,. continuous for a certain value x. ,. of x,. quantity, and A?/ approaches zero as positive or negative.. The. latter condition is. sometimes expressed, "when an infinitely. small increment in x produces an infinitely small increment in y.". The most common. case of discontinuity of the elementary functions. (algebraic, exponential, logarithmic, trigonometric. nometric, functions). is. when. the function. and inverse. trigo-. is infinite.. Y. ^. a. "-——. A. °. -. \ For example, consider the function y for all values of x except x = a.. When taking x. x. = a, y =. when x>a, y There. oo,. that. is,. sufficiently near a.. is. =. 1. which. is. continuous. y can be made as great as we please by Also when x<a, y is negative, and. is positive.. a break in the curve. when x. to be discontinuous for the value x. = a.. = a,. and the function. is. said.

(39) •. .. DERIVATIVE. The function y. (x-ay. is. 23. likewise discontinuous. This function being positive for. all. values of. x,. when x = a.. the -two branches. of the curve are above the axis of x.. Likewise the functions, tan. In general,. if/(.r). = oo,. y=f(x) corresponding. x,. sec. x,. when x = a,. to x. = a, and. are discontinuous. there. when x = —. Z a break in the curve. is. both the curve and the function. are then discontinuous.. I X. Another form of discontinuity. is. seen. 2 in the function y = —. when a = 0. limits, according. Lim z = +. -^—. =1.. see that. when x =. curve jumps from. from. B to. The function ous for x. y=2. to. the. y=l,. A. is. discontinu-. = 0.. It is to be noticed that the. definition. of. the. Lim. i. r+i. 2'+l. is. ,. as x approaches zero. through positive or negative values.. that. 2. 2*+l. Here y approaches two. We. 4-. derivative. = 2..

(40) DIFFERENTIAL CALCULUS. 24. implies the continuity of the function. limit, unless. Ay approaches. The converse. zero. For —^ cannot approach a. Ax when Ax approaches. zero.. There are continuous functions which have no derivative, but they are never met with in ordinary not true.. is. practice.. EXAMPLES The equation. 1.. Find. (a). of a curve is y. its. Find where. points. curve. parallel. is. + 2.. inclination to the axis of. when x = (b). x>. x,. when x = 0, and 0°. Ans.. 1.. and. 135°.. the. the to. the axis of X.. x=0. Ans.. and x=2. Find the where the. (c). points slope. unity.. is. Ans.. = (1±V2).. £C. (d). Find. the. point where the direction. is. the same as. that at x > 2.. = 3.. Ans. x. In Problem. 1,. when man ?. Art. 20,. be the same as that of the. When. = — l. will the velocity of the. Ans.. one quarter, and when nine times, that of the Ans.. ^3.. A. When. AP = 12. circular metal plate, radius r inches,. the radius being expanded. m. At what. rate is the. conditions of Ex. 3 ?. is. inches per second.. the area expanded ? 4.. When. ft.. man ? ft.,. and 32. ft.. expanded by heat, At what rate is. Ans. 2 irrm. volume. shadow. AP= 24. sq. in.. of a sphere increasing. Ans. 4 Trrra cu.. per. sec.. under the in.. per sec..

(41) DERIVATIVE 5.. The radius. of a spherical soap bubble is increasing uniformly. at the rate of yL inch per second.. volume. is. increasing. when. Find the rate. the diameter. Ans.. 6.. In Exs. ". Is. 7.. »+l. 5, 7,. is. at. which the. 3 inches.. ^ = 2.827. cu. in.. per. sec.. Art. 16, is y an increasing or a decreasing function ?. an increasing or a decreasing o function of x ?. In the Example. ing function of 8.. 25. x,. and. 1,. for. above, for what values of x is y an increaswhat values a decreasing function ?. Find where the rate of change of the ordinate of the curve + o, is equal to the rate of change of the slope of. s y = X — 6.i~ + 3.r the curve. Ans.. x= 5. or 1.. — 3. 9.. When. is. the fraction. —-x x +a 2. -. 2. increasing at the same rate as. Ans.. When. x2. a??. =a. 2 .. 10. If a body fall freely from rest in a vacuum, the distance through which it falls is approximately s = 16 t 2 where s is in feet, and t in seconds. Find the velocity and acceleration. What is the velocity after 1 second ? After 4 seconds ? After 10 seconds ? ,. Ans. 32, 128, and 320. ft.. per. sec..

(42) CHAPTER. III. DIFFERENTIATION 23.. The process. of finding the derivative of a given function is. called differentiation. trate the. meaning. The examples. in the preceding chapter illus-. method of any but the. of the derivative, but the elementary. differentiation there used. becomes very laborious. for. simplest functions. Differentiation. is. more readily performed by means of certain. general rules or formulae expressing the derivatives of the standard functions.. In these formulae u and v will denote variable quantities, funcand c and n constant quantities.. tions of x. ;. It is frequently convenient to write the derivative of a quantity u,. — u instead of — dx dx the symbol. Thus. 24.. A^ ~r dx. v) ^. t. — dx. denoting " derivative of .". k e derivative of (u. -f-. v\. may be written. Formulae for Differentiation of Algebraic Functions.. i.. ^=1. dx. II.. TTT. — = dx. 0.. — dx. 4-. (. .. }. —— dx. 26. — dx. -l-. — dx. (u-\- v),.

(43) •. DIFFERENTIATION du d — («r) = v dx dx. TTT. f. IV.. d dx. V. N. ^ __. ,. vi. h. —. ?<. dx. du dx u. dx. dx .. v2. dx\vj VII.. —. dv. .. V 7. A/ ^—. 27. (vr)=nun ~ 1 —.. dx. dx. These formulae express the following general rules of ation. differenti-. :. I.. TJie derivative. II.. TJie derivative. of a variable with respect of a constant is zero.. III.. TJie derivative. of the. sum of two. of. product of two variables. to itself is unity.. variables is the. sum of. their. derivatives.. IV.. TJie derivative. the products. tJie. of. tJie. eacJi variable. by. tJie. is tJie. sum of. derivative of the other.. TJie derivative of tJie product of a constant and a variable is V. product of the constant and.tJie derivative oftJie variable. VI. TJie derivative of a fraction is tJie derivative of tJie numerator. multiplied. by. tJie. nator multiplied by. denominator minus tJie. tJie derivative of tJie denominumerator, this difference being divided by the. square of'the denominator. VII. TJie derivative of any power of a variable exponent,. tJie. power. exponent diminisJied by. witJi. is tJie. product of the. &nd. the derivative. 1,. of the variable. 25.. Proof of. a derivative.. 26.. I.. For, since. Proof of II.. vary.. Hence Ac. This follows immediately from the definition of. =. and. A. \w — = Ax. 1, its limit. constant. — = Ax. j. is. 1.. a quantity whose value does not. therefore •. dx — = dx. its. limit. — = dx. 0..

(44) DIFFERENTIAL CALCULUS. 28. Let y. Proof of III.. 27.. = u-\-v,. and suppose that when x reAu and Av,. ceives the increment Ax, u and v receive the increments respectively.. Then the new value y. therefore. Divide by Ax. of y,. + Ay = u + Au + v + Av, Ay = Au-{- Av.. then. ;. Ay _Au Ax Ax. Now. suppose. Ax. Av Ax. and approach. to diminish. and we have for. zero,. the limits of these fractions,. dy _du dx dx If in this. we. substitute for. u. y,. —. d / (u dx. dv dx' -{-v,. we have. + v) = du N. .. 28.. and. We. ,. s. ,. ,. du dx. .. dv dx. should then have ,. = uv; y + Ay = (u + Au) (y + Av), Ay = (u + Au) (y + Av) — uv = vAu + (u. Divide by. Ax. dw. ,. dx. -f-. Au) Av.. ;. ^/ = „^ + M + Au) ^. Ax Ax ( Aaj. then. Now u + Ait. ,i. that. .. dx. Let y. Proof of IV.. then. dv. 1. same proof would apply to any number of. It is evident that the. terms connected by plus or minus signs.. d , dx. ,. dx. suppose Ax to approach is u,. dy -£. = v du. dx. dx. —d (uv)=v du dx /. is,. zero, and, noticing that the limit of. we have. dx. k. ,. \-u. ,. \-u. —. dv dx dv — dx. ;.

(45) \. a. .. DIFFERENTIATION Formula IV. may be extended Thus we have. Product of Several Factors.. 29.. 29 to. the product of three or more factors.. —. d , v d , d , dw x as (uv -w)=w (uv)J + UV ( uviv) y K K ) } dx dx dx dx. —. —. •.. = w[ v du. ,. ,. \-u. = vw du. —. dv\. .. -{-uv. ]. ,. \-uw. dx. dw dx. dx. dx. dv. ,. [-uv. dx. dw — dx. from the preceding that the derivative of the product two or three factors may be obtained by multiplying the derivative of each factor by all the others and adding the results. This rule applies to the product of any number of factors. To prove this, w& assume It appears. of. —d. ( I. Wh. •••. Then ^-(. u n \j. l(. \. u. -2. =u. 2. u3. un. •••. —. du, -f. u u z uA. .... x. — w w n+1 = un+l -—( UjU B. 2. un. ••. •. —. du„ -. h u x u 2 ~-u,. uA + u&v. •••. un. J. =u u 2. +. ?,. Thus 7i-|-l. lW2. 3. • • •. .... un+1. — + Uju u. du*. duo. ,. i. 3. ax. 4. • • •. u n+l —? H ax .. p.. du n dx. .. H. w^. • • .. du„^ dx. w n _!W n+1. du — dx. n. Wn _n±i.. ax. it. appears that. factors,. and. if. is. the rule applies to n factors,. it. holds also for. consequently applicable to any number of. factors. Tlie derivative. of the product of any number of factors. is the. sum of. the products obtained by multijjlying the derivative oj each factor by all the other factors..

(46) DIFFERENTIAL CALCULUS. 30. 30.. This. Proof of V.. we may. derive. it. is. a special case of IV.,. dc — being dx. zero.. But. independently thus. = cu,. y y. + Ay = c(u + Aw), Ay = cAw, Aw Aw — = — Ax Ax c. dy -^ dx. 31.. = c du —. y. x. c. du dx. =u. + Aw. w. + Av. v+Av. p. '. v. Aw. w. Av. Aw_ ^ x ^x Ax~ (y + Av)v. and. Now. ,. dx. Ay = «±* tt - ? = »* M ~ "f ». therefore. v. or. ~Lety. Proof of VI.. Then. — (cw) = d,. ,. dx. Ax to approach we have. suppose. + Av is. v,. dy. dx. zero,. _. V. and noticing that the limit. dw dx. ~~. u v2. Or we may derive VI. from IV. thus Since. y. = -, v. therefore. yv. = u.. dv dx. of.

(47) ,. .. DIFFERENTIATION t> By. dy. ttt. v-f + dx. IV.,. .. dv du =— y—. dx. dx. dy __ du dx dx v. _. dy. therefore. u dv, v dx. du dx. dx Proof of VII.. 32.. y. and. that. 2. = un + Ay = (u + Aw) n Ay = (u + A?*)' — ,. ,. 1. Putting. Ay. —. dv. dx v. y. then. u. 1. suppose n to be a positive integer.. First,. Let. 31. =. u". .. u + A?<, we have = (V — u) (u" ~ + u ~ u + ~3 w H Ay = Au (u'*- + it" - u + u*^V w*= (u*- +u' u + m'- ^ + a- —. u' for. —. un. l. l. 2. hl. «4. 1. is,. 2. 1. let. 1. n. 2. 2. 8. 1 ). .... then, u being the limit of. terms within the parenthesis becomes u n -/. ax. = nun ~. l. ~l ;. —. u',. each of the n. therefore. •. dx of Art. 29,. ,. — (O =u — +u n ~ lC. dx. du dx. -1. n. dx. +. .... to. dx Second, suppose n to be a positive fraction,-?.. Let. y. then therefore. if. = u", = up. ;. — Of) = — (u dx dx. p. KJ. J. ),. Ax. Or it may be proved by regarding this as a special case where u l7 u 2 ••• and u n are each equal to u.. Then. ~l. ),. '. ;. un. 1. N. Ax diminish. +. 2. ,B. f-. ^ Ax Now. un. K. ). }. n terms.

(48) DIFFERENTIAL CALCULUS. 32. But we have already shown VII. a positive integer equation.. ;. hence we. may. to be true. apply. when. the exponent. member. to each. it. This gives qyq ijL. =pvP. i. .. dx. dx. 4=«!^* ~. therefore. dx. q yq. l. dx. 2. Substituting for. y,. uq gives ,. dy dx. which shows VII. Third, suppose. by. is. u. _p. P. q~ l. q. du dx'. q. to. —. m.. = u~ m =: —m u. —=. dx su 2m. dy /- = dx. Hence, VII.. du q p _pdx. n to be negative and equal y. T7T VI.,. l. to be true in this case also.. Let. i. _p up ~. —. dx. u 2m. — = - mic m - ,du 1. dx. true in this case also.. EXAMPLES Differentiate the following functions 1.. y. = x\. * = !(*•). dx dx If. we apply. VII., substituting. u = x and w. — (z )=4ar — =4ar>. dx 4. dx Hence,. K. }. J. ^ = 4^. dx. by J. = 4,. I.. we have. is. of this.

(49) a. DIFFERENTIATION 2.. y. = 3» + 4aj 4. 8 .. *? = J*(3a;*+4a?)=. by. III.,. making. 33. u. = 3x. 4. and. u. = 4#. — (3. +-^(4. x*). 8. ),. 3 .. byV.. |(3^)=3|(A. = 3-4^ = 12^. = 4 — (or) = 4. Similarly,. —. Hence,. ^= 12a. (4. x3 ). 3. .. 3 x2. 12«a = 12 (a?. +. = 12 a?.. +. 2. a?. ).. e2as. 3.. y = ** + 2. dx. dx. |(,f). dx. = |,i. by. |(2)=0, Hence,. 4.. ;. by VII.. II.. ^ = 1** dx 2. y=3Vx-A + | +a dx. dx. .. y. J. ^. 3 -I. = 2*. -. K. 2 (-_1. --*- + _1 _ 2x*. dx. 3 '. x4. J. dx. K. '. dx.

(50) DIFFERENTIAL CALCULUS. 34. 5.. y. = %£ ar + 3 dy_. d. /. + 3\. x. dx\x 2. dx. -{-3. Applying VI., making u=x. +. (-. dfx+3\ <toV<s + 3y 2. 2. 3 and v. = x + 3, 2. we have. + 3)|(, + 3)-(, + 3)|(^ + 3) (x + 3f 2. 3-6a;-^ _ f + 3-(x + 3)2x = ~~ (z + 3) (> + 3) 2. dy = dx. Hence,. 9. 6.. S-6x-a* (x + 3). m. = (x + 2)t. 6. 2/. we apply. dx. making. VII.,. ^. =x +2 2. and w. = 2-,we. 3(»2. Hence, dx. y. have. o. 3. 7.. 2. 2. 2. dx If. 2. 2. 3(x2. + 2)3. = (x + l)Vx*-x. 2. ^=A[(aj». + l)(aJ»-aOH. + 2)i.

(51) ,. DIFFERENTIATION If. we apply. IV., m. =. 35. making as?. +1. and v. = (x — x)-, 3. we have. £[(rf + i)(#-.)»]. =. dx. + 1) |- (X - »)* + (*" - K)i|s. (x2. ^= =. i(.T 2. (j. 2. + 1)(3 x - 1) (ar. + 1) (3. 8. 10.. y. = (x + 2a)(x-a). ,. s,= (»*-ai)«. '. 13.. + ar -5, ?. ,. Example 11. ~ v= 2x 1 (x-if. J. y. = z(a*+5)*,. =7. -»)*2a. -2 -1 2{x - a>)* 4. 2. a;. a;. z. ^ = 3 (10 a. -4z. 9. ^=3 (a -a 2. 2. ,11... Differentiate. a;). (s3. 3. = 3x. -2a;6. -*)-£ +. - 1) + 4 afo (.r - xft. 2 x-. ^. 10. 3. 2. 8.. 12. + 1).. + l)=2x.. 2. 11.. 2. dx. 2. dx. Hence. 2 (a-. <fy. S=. 4. (a; 3. —. 5. 2. ).. q3)3. 3ic f. also after expanding.. dy = dx. g=5. 2x (x-iy (ar. + l)(x + 5)i 3. 2. -far )-.

(52) DIFFERENTIAL CALCULUS 14.. y. '. Va - x 2. 2. (^-a )^. —x. la. 16. dy dx. x. members. Differentiate both. 18.. /'^-±A Y. 2. a=. 21. ^^ y y. * (*. (. 2. = ^ + 2a + a »2. ^. 2. ~ 3). + 2). f=. 25.. 2. + a )(* + a )T 2. 2. 2. 2. 3. n-i. aJ_a. (?. dx. '. + l) (3z-8). V2 aa; -. (nt. dt. = *(*» + n)->. </=. .. <fy_6(3* +4Q(2; 2 -3) 2 '. (2a-3a) 9. N. <,. 4. 2. 4. (z. (2a-3a?) 10. 2. -. oj. 24) (a;. + 1). 2. (3. dy_ n(xn + l). ^d2/. 2. + 2f. + 2^. ^ = 3 (13 a 24.. 2. .. 3. 2. —x. 2. + <*)*&. 2. (?. 2,= (z. 4. ^_ = 1+. 2 ,. 20.. 23.. 2 xy/ax. ). 2. *. 2. of the identical equations, Exs. 17-19.. 2. 2. (x 2. '. -xf. + ax -h a )(x — ax + a = x* -f aV + a. 17.. 19.. (a. 2. dy = 3aWx*-a 2 dx. 2. 15. a2. dy dx. x. =. *». (ajn. + n) l a'. (2. aa - x2)%. a;. - 8). 3. (x. + 2). 5 ..

(53) x. 1. 1. DIFFERENTIATION. *"•. *. ~~. '. ~4. 37. 4 ^.4 Q x 3. ,7^ x rf. cty. 28.. g= d* 4. *=(*^2)JJ7i; \. 29.. 30.. t,. (x3. _. — «?)». if. c7y. l'. y V. = ^jX-. 6 x2 4- 6 *. 31.. +. (.r. =. (x. + Vx. 2. + 8az +. n. 4- l) (". 15«. For -what values of x Arts.. A. 2. )3 ). - 9a ) _ 3 cu.) 5(4 rf + 8 ax + 15 a )* (rf 4(2. 3. 3. 2. 2. te. 34.. 9. or. Vx + 1 - *), cZ.V. ing function of x. 2. 4. (4« + l) f. *b y. +x + 1) Vx + x +. 12. c7?/_. 33.. )V-«') f x-'-l. *. 2/== ( iC2_3ax)^(4^. 3_h a3. (a*. dy_. (4*+l) f 32.. 2 a 3x 2. cfy_ ax. ~X-\-l. t. + 1)1. _. d*. \^+x + l'. 3 (ti. is. = (,r-l)(z+Vz + l)».. 3 x4. 2. —8x. 3. an increasing or a decreas-. ?. Increasing,. when x. >2. ;. decreasing,. when x. <. 2. form of an inverted circular cone of semiwith water at the uniform rate of one cubic foot per minute. At what rate is the surface of the water rising when the depth is 6 inches ? when 1 foot ? when 2 feet ? 35.. vessel in the. vertical angle 30°, is being filled. Arts.. .76 in.. ;. .19 in.. ;. .05 in., per sec..

(54) DIFFERENTIAL CALCULUS. 38 36.. The. side of an equilateral triangle. of 10 feet per minute,. How. second. 37. vessel, hour'.. and the area. is. increasing at the rate. 10 square feet per Ans. Side = 69.28 ft.. at the rate of. large is the triangle ?. A. Another vessel is sailing due north 20 miles per hour. 40 miles north of the first, is sailing due east 15 miles per At what rate are they approaching each other after one After 2 hours ? Ans. Approaching 7 mi. per hr. separating. hour ? 15 mi. per. When. ;. hr.. will they cease to. approach each other, and what. is. then. their distance apart ?. After 1 hr. 16 min. 48. Ans. 38.. A train. starts at. being represented by. From Worcester,. f.. = 24. Distance. noon from Boston, moving west,. =9. s. sec.. its. mi.. motion. forty miles west of. Boston, another train starts at the same time, moving in the same. motion represented by s' = 2 f. The quantities s, s', and t in hours. When will the trains be nearest together, and what is then their distance apart ? Ans. 3 p.m., and 13 mi. direction, its. are in miles,. When. will the accelerations be equal ?. Ans.. =. If a point moves so that s y7, show that the acceleration negative and proportional to the cube of the velocity. How is. 39. is. 1 hr. 30 min., p.m.. the sign of the acceleration interpreted ? 40.. Given. 41.. A. = - + bt. s. body. starts. the coordinates of. its. from the. 2 ;. find the velocity. origin,. and. acceleration.. and moves so that in. t. seconds. position are. o. Find the. rates of increase of. Also find the velocity in ds. its. x and path,. vdHty vn. ,. ,. y.. which. is. Ans.. 5f + o..

(55) DIFFERENTIATION 42.. axis of. Two. bodies move, one on the axis of. and. y,. At what 1 minute ?. in. t. x. 5=. 39. and the other on the. aj,. minutes their distances from the origin are 2. 2. f. -. 6. t. and y = 6 1 — 9. feet,. feet.. rate are they approaching each other or separating, after. After 3 minutes. Ans.. ?. Approaching 2. ft.. per min.. \Yhen will they be nearest together. separating 6. ;. per min.. ft.. Ans. After 1 min. 30. ?. sec.. M. are the middle points of BC 43. In the triangle ABC, L and and CA respectively. A man walks along the median AL at a uniform rate. A lamp at B casts his shadow on the side AC. Show 2 3 2 4 2 and that the velocities of the shadow at A, M, C, are as 2 3 3 3 3 4 that the accelerations at these points are as 2 :. :. :. Suggestion. line parallel to. 33. Formulae. — P being. :. ;. .. any position of the man, draw from. L. a. BP. for. and Exponential. Logarithmic. Differentiation of. Functions.. du VIII.. -^-log u «. = log e^. a. dx. TV IX.. d. du dx. — log w=—u i. — a dx. u. *e". XI.. = log a-au — e. =e. v. d -. dx. •. dx. — dx. dx. YTi XII.. •. e. dx. X.. u. "='//. .du. dx. ,. -,. h log e. ?^. •. uv. dv. —. dx. •.

(56) DIFFERENTIAL CALCULUS. 40 34.. Lety = loga u,. Proof of VIII.. then. y. + Ay = \og. a. (u. + Au),. Ay = \og a (u + Aw) - log a u = lo go. ^±^. = log/l + ^=^log„(l+^f. ".. u J. \_. u. \. u J. Dividing by Ax,. ^logA+^f^ u J u. Ax If. Ax approach. Au. zero,. \. likewise approaches zero.. Now Lim^^l + ^Y^Lim^Jl + l For,. if. — = Au. we put. i+ (. and as An approaches. But. in Art. 12. fr=(i+ zero, z. Hence,. we. infinity.. we have found. Lim AM=0 [ 1 if. t:. approaches. Lim^M +-j therefore. z,. —. -\. =e;. - r" =. e.. take the limit of each. dy dx. i. member du dx u. of (1),. CD.

(57) .. .. DIFFERENTIATION This. 35. Proof of IX.. 41. a special case of VIII.,. is. when a = e.. In this case log a e. Note.. — Logarithms. Hereafter,. e. to base e are. when no base. understood; that. = log e = l.. is specified,. called Napierian logarithms.. Napierian logarithms are to be. is,. log u denotes log e 36.. u.. Proof of X.. Let. = au. y. .. Taking the logarithm of each member, we have log y. dy dx. = u log a;. — — = loga du dx. therefore by J IX.,7. ,. y. Multiplying by y. = au. ,. we have dy — = loga dx t. 37.. Proof of XI.. 38.. Proof of XII.. This. is. •. au. du — dx. a special case of X., where a. Let y = u v. —. e.. .. Taking the logarithm of each member, we have. \ogy=v\ogu; therefore. dy dx. by J IX.,. —=. '. y Multiplying by y. =u. v ,. du dx do + logw— dx u .. n. we have. —. dv dy — r-idu vuv l hlog^-^ v dx dx dx .. -?-. The method each member,. of proving X.. may. This exercise. i. and XII. by taking the logarithm of. be applied to IV., VI., and. is left. to the student.. VII.

(58) DIFFERENTIAL CALCULUS. 42. EXAMPLES (See note, Art. 35.) 1.. y. = log. 2.. y. = xn log (a# + 6),. 3-. 4.. y. 2/. (2^. + 3 r%. 10. (3. ox. dy dx. ax+b. dy _ _ dx. xlogx. = log. dy _ 6 (a; + 1) dx 2 a? 2 + 3 x. x. + 2),. riy. da. 1. + log x. (x log x) 2. = 3 log e = 1.3029 3x + 2 3a+2 10. 5.. #. = log ax — b aa? + 6. dy _ 2 ab dx a2x'2 — b 2. I.. y. = lo{ 3^ + 1 «+3. dy dx. '. dy eft. 8.. y. =ae. 9.. y. = log(a* + 6*),. 10.. y. +nlog(ax+b). 3a + 10a;-f 3 2. _. 2. (f-. - 1). ^4-^-j_^. |2=(l + loga)a*e".. x x ,. dy _ a x log a + ~ dx ax + dy =8 dx. = (««-_ 1)*,. e. 2x. (e. 2x. &*. b. log b. x. - l). s .. Differentiate Ex. 10 also after expanding.. 11.. 12.. y. V. = 6* + x—3. y=(3x-l). <fy. »,. cte. 2 3x. e. -2 ,. dy dx. = 24a--10^. &. (x-S) 2. 3 (9 x2. -. l)e 3x -\.

(59) v. DIFFERENTIATION 13.. y. =x. 5. 5x. ~JL ,. =x. 4. 43. o T (o. +x. log. 5).. da;. 14.. = log. y. d# dx. 1. log x -. log x. members. Differentiate both. +e. 15.. (x. 16.. (a*. 17.. log. 18.. .i. ... 20.. 4 .r. x. 2 (e *. + 4 .rV + 6 xV" +. +. 2a e'. ). = log. x. (e. ~a. 3 a x e 2x. +e. a"x. ). 4 a* 3*. +e. + x + a.. .. x. /. i. (Vx~HTa. ,. +. dy. ^. .. Vx),. log x. dx dy _ da. 22.. y. = log. ^±i-^ + +. dy _ ax. 1. = x [(log x) - 2 2. 1. log x. + 2],. 23.. y. oa 24.. r = log(VJ+7-V—,fc. 26. ,,. i. /. V4r>- 1 1. xVx + 1. 4 = (log X) UX "'•'. /. S. ,r '.-"'. - 1 + *'"' + 1 + <r". •''! ,. dx. ax. 2. 2-. — ^ -. +. (x. a)*. + (x - a)* ..'. .. = log(V* + 3 + V* + 2)+V(* + 3)(z + = W-. +. 2Vx*. = log (2a-+V4x*-l),. a.,-. 1. <ty_. y. 25.. 4 *.. - e*.. 21.. i. a;). log x. los:. = log. x. ". =a. x. y. y=. 2. x (log. of the identical equations, Exs. 15-18.. - e f = a3 - 3 cr*e +. Jog a. Q. x. _ 1 + log X. 2),. f. g=^/|±|. = 2 „(,'-?-<) c-"' + 1 + e"-".

(60) DIFFERENTIAL CALCULUS. 44. 87.. lo« x. y = log 1. +. ®L =. ,. log x. ,. x log. dv. 30. + log x). (1. a;. .. e°*. = 3 log ( Vx + 3 -3) + log ( Vx + 3 + 1), 2. 2. 2/. 1. dx. 4. fly __. a;. ^"~^-2V^T3 30.. y. = log. (a. + V2ax-a ) + 2. Va 4- V2 — a a;. 0> dx. + l4-V^ + 3a + l 2. a;2. v _,i og. 31. ;. j. x. 1. 2(x. + V2 ax - a ) 2. dy = x2 -! dx x^/x^ + Stf + l. The following may be derived by XII. or by differentiating after taking the logarithm of each member of the given equation. ^| = nx™. 33.. y. = xnx. 34.. y. = (ax>) x. 35.. 2/. = x**. 36.. 2/. = (logx)*,. 37.. y. = x<. 38. ,»(-£_£ \x. .. ,. ,. 2. *. ^. + aJ. 1. 2. ). + log x).. [2. + log (ax )]. 2. = + 21ogx). ^ ax = ° x)/-i— + log log x V ^ ax Vlogx y * (w + V <^ ax°*. ,. 10. (ax ^= ax. (1. 2+1. (l. (log. te. * dx. rf. 1} (log. ,,). io,. f-fL,W-i-+il€g^-A. x + a) a \x-\-aJ \x + a.

(61) .. DIFFERENTIATION The method expression. of differentiating after taking the logarithm of the. may. This. tions.. 45. is. often be applied with advantage to algebraic funcsometimes called logarithmic differentiation.. In this way differentiate Exs. 21-26, pp. 36, 37. I. 39.. Find the slope of the catenary y = ~(e a. What. is. When. X?. x=. at. ),. 0.. inclined 45°. is. Ans. x = a log e (1. + V2).. does log 10 # increase at the same rate as #?. When. Ans.. When. a. the abscissa of the point where the curve. to the axis of. 40.. X. e. -f-. at one third the rate?. x. = log. When. Ans.. 10. e. = .4343.. x = 1.3029.. Verify these results from logarithm tables. 41.. show that the acceleration 42.. by a point. If the space described. If a point. is. is. given by. s. = ae + be~ c. moves so that in. seconds. t. s. = 10. log. feet,. and acceleration at the end of 1 second. At the end = — 2 ft., and — .5 ft. per sec. Acceleration = .4, and .025.. - 2) - 9 ^~ 36 x + 32 ~ ) \ x > 3; decreasing when x < 3.. y = log (x an increasing or a decreasing function ?. For what values of. a?. is. Ans. Increasing when 39.. +4. Ans. Velocity. of 16 seconds.. 43.. ,. equal to the space passed over.. £. find the velocity. f. Formulae. for. Differentiation. Trigonometric Functions.. of. the following formulae the angle u. 3. is. circular measure.. XIII. d du — sin w = cos u—dx dx. XIV. —cos u =. XV.. dx. - sin u—. tanw = sec — dx. dx. 2. In. supposed to be expressed in. M— dx. ..

(62) DIFFERENTIAL CALCULUS. 46. du d cot u = — cosec w — — dx dx o 2. ,. XVI. see u= — dx. XVII.. u tan %. sec. •. — dx. — — cosec u = — cosec u cot dx. XVIII.. ?t. -. dx. —-versw=sm?t—. XIX.. ax. ax. 40.. Proof of XIII.. then. = sin u, y-\-Ay = sin + Am); A y = sin (w A u) — sin Let y. (it. therefore. ?t.. -f-. But from Trigonometry, sin. If. we. A - sin B = 2 sin* (A - B) cos. substitute. ^4. = u + A w,. and. \. (A. B = m,. —-)sin—. Av = 2cos(mH. we have. + B).. -. 9. Aw Hence,. Now when Ax as. Au. is. = H Ax. u. cos. + *g) 2. V. approaches zero,. Aw. likewise approaches zero, and. in circular measure, •. sin. Au 2. Lim AM=0 -^-. = 1.. T TT. l^. An Ax. Hence,. dy — = cos a du. dx. dx. See Art. 12..

(63) — DIFFERENTIATION 41. Proof of XIV. for u,. may. This. 47. be derived by substituting in XIII.. £-?/.. Then. d cos?* dx. —. or. Proof of XV.. 42.. = cos(|-„)|(| r «). |-(l-«). —. du\ dxj. f. = sin. ?/[. V. =-sm« d». dx. = ^-^,. Since tan u. cos u. cos. „T. .. bv. tf. ~y~. \ 1.,. d*. tan ". — sm tf. ?/. .. v. — sm. d — cos u. ?/. (to. ete. =. 5. cos-. >/. dv dn b— t sm- u —. ,. cos 2. .. .. COS". du —. o. COS". ?'. w.. sec- " CfeB. ing. This 'may be derived from XV. by substitut-. Proof of XVI.. 43.. ——. v for. ?/.. 44. Proof of XVII.. Since sec u. =. — cos u. d ,. „,. f?. 3~ sec. bv J A L,. ax. 7. u. .. =. =. 5. cos-. = sec. sm. cos u. eta. =. cos-. ?<. c?u. dx u. — du. .. v tan. w. >/. dx 45.. Proof of XVIII.. stituting. ^. — u for. be derived from XVII. by sub-. u.. 46. Proof of XIX. relation. may. This. This vers. is. readily obtained from. u= 1 — cos 1. u.. XIV. by. the.

(64) DIFFERENTIAL CALCULUS. 48. EXAMPLES y u. =3. 2.. y. = log cos. 3.. ?/. = log. 1.. sin. — 2 cos. 3x cos 2x. Sx sin. = 5 cos 3x cos. 2x, -^. 2x.. dx 2. x-\-2x tan. a?. —. 2 a?. -^-= 2x tan 2. ,. —= m. (sec ra# -f tan mx),. #.. sec wa?.. C*3J. ?/ u. = log5. (a v. y. = cos. a log sec. i. >i. 4.. c 5.. 6.. /. sm •. 2. + 6* i. cc. //). i. y=(m — 1). (8. sec m+1. dy tan # — = 2—(a—b) r dx a tan x +. \ cos 2 a), ;. 1. 2. — a) + a sin \. dv. •. ot,. sin. —. =. -^ dv. ,. b. cos (0. — a). .. #— (ra + l)sec m_1 a;,^==(m — l)secm-1 a;tan 2. 3 a;.. cia;. 7.. 2/. 8.. r. 9.. = log tan = log&. (. aa?. — ^"j,. tan 6 (sec v. [sec L. =— 2a sec 2ax.. -^. + tan. 2. 0)y J ]. , '. — = (sec 6 + tan ^ dB. 2 -. tan. = cosecm ax cosec" bx, —^ = — cosec™ ax cosec" bx (ma cot ax + n& cot. 2/. bx).. dx. 10.. = 2x. w. 2. sin. 2x. + 2x cos 2# — sin 2x,. -&. ==. 4a2 cos. 2a\. da;. 11.. =2. ?/. tan3. a;. sec. a;. + tan. a;. sec x. — log. + tan #), -^ = 8 tan x sec. (sec. x. 2. 3. a;.. dx. »- sin ?-f cos x '. -. 12 -. 13.. y. dy. a;. '. = e '(sin 2a; - 5 cos 2a;), 3. V. 2 sin x. Tx. ^ =13e. ax. to. (sin2x. - cos2z)..

(65) «. DIFFERENTIATION 14.. y. +. cos (x. 15.. ,. 16.. 17.. y. = sin. y. = log. 2/. 3. t. =. dy dx. cos x. = log. a). sin. x. 4- vers. sin. x. — vers. !. (sin 2. -^ dx. cos 4 3x,. 4.r. = 12. x. sin a. + a). cos x cos (x. sin 2 4a; cos 3 3a; cos 7x.. dy -^. ,. = secx.. dx. a;. —=. 1 a;). 49. ,. 2/. (log sin. + 2a;. 2x. cot. 2a;).. da;. 18.. 19.. 20.. = (tan. ?/. y. a;). = (sin x). 7/. — = y (cos. 8inz ,. log c0 * x. -^. ,. dx. 7/. = (tan -3 cot a;. sin^fl 22.. y. 23.. 24.. y. y. a;). sin. a;. Vtana;,. — «). a;. + sec. - = 2 sec. 3. x.. da;. dy d*. = 3sec. dy. _. 4 a;. 2 tan* x sine*. d0~"cos«-cos0. + a)'. = a log (a sin + 6 cos. a;). + 6a,. ^ = _^_±&!_, da;. a tan. da;. 1-|- sin 4a;. dy d*. 4-5 sW. a;. -f b. =2-. sin. (. + i). tan25.. a;. a;. = log sini^. log tan. -2. y = log. 2tan?-l. a?).. = y (cot x log cos — tan x log sin x).. = tana;seca; + log J^L±4^, — * 1. 21.. a;. dx. 3.

(66) DIFFERENTIAL CALCULUS. 50 oa «D.. = a sin + b vers x a sm x — b vers x. 2 ab vers x. dy — - —. if. ;. ]). ,. dx. —. ;. (a. sm x — b. In each of the following pairs of equations derive by two equations from the other:. vers x)~.. differentia-. tion each of the. 27. .. = 2 sin x cos x, — sin cos 2 x = cos sin 2. a;. 2. 28.. sin 2. 2. a?. ic.. 2 tan x a;. 1 -f tan 2 aj'. —. 1. cos 2. tan. 2. aj. a*. tan 2 # 29.. = 3 sin — 4 sin — 4 cos — 3 cos #. cos 3 x sin 3. 3. a;. a?. 3. 30.. sin 4. cos 4 31.. 32.. onds,. a?. a:,. a;. = 4 sin cos — 4 cos = 1 — 8 sin x cos x. 3. a;. a:. (m + n) x =. 33.. a;. sin 3 #,. 2. mx cos + cos mx sin wa;, cos (m -\-n)x= cos mx cos wa; — sin mx sin wa\ sin. sin. rase. made in ir when = 0°,. If 6 vary uniformly, so that one revolution is. show that the. rates of. increase of. 45°, 60°, 90°, are respectively 2,. of tan. a;. 2. V3, ^/%. when. = 0°,. 1. 0,. 0,. 30°, 45°, 60°, 90°,. sec-. 30°,. per second.. show that the. If 6 is increasing uniformly, 0,. sin. rates of increase. are in harmonical progres-. sion. 34.. For what values of. 0,. less. than 90°,. is. sin 6 -f cos. an increas-. ing or a decreasing function ?. Find. its rate. of change. when. =. 15°.. The crank and connecting rod. Ans.. vr. and 10 35. feet respectively, and the crank revolves uniformly, making two At what rate is the piston moving, when revolutions per second. of a steam engine are 3.

(67) DIFFERENTIATION makes with the. the crank. line of. 51. motion of the piston. 0°,.. 45°, 90°,. 135°, 180° :. If a,. and. the triangle,. are the three sides of. b, x,. opposite. $ the. angle. b,. =a. x. + V6 — a2. cos. Ans.. A. OP PQ. 0,. sin2. 6.. 32.38, 37.70, 20.90, 0,. ft.. per. sec.. O with angular velocity <o, and a hinged to it at P, whilst Q is constrained to Prove that the velocity of Q is w. OP, move in a fixed groove where R is the point in which the line QP (produced if necessary) 36.. crank. connecting rod. revolves about is. OX. meets a perpendicular to. OX drawn through. The inverse trigonometric. Inverse Trigonometric Functions.. 47.. functions are many-valued functions; that x.. there are an infinite. For example, sin. But. -1. the angle. if. number. is,. where n. any given value of. for. of values of sin. -= ±-± 2mr, is. O.. -1. is. #,. any. tan. -1. #,. &c.. integer.. restricted to values not greater numerically. than a right angle, sin -1 x will have only one value for a given value of. x.. cosec. the. Then -1. x,. first. sin. tan. -1. -1. x,. -. = -,. 2. 6'. and. cot. sin. _1. _1 .T,. =—. 6 2y V as taken between (. ]. •. We —-. thus regard sin. and. -, that. -1. is,. x,. in. or fourth quadrants.. But cos _1 .r,. -1. and vers -1 a;, must be taken between and v, that is, in the first and second quadrants, which include all values of the cosine, secant, and versine. These restrictions are assumed in the following formulae of differsec. x,. entiation.. 48. Formulae for Differentiation of Inverse Trigonometric Functions. flu. XX.. — sin. VI - u2. dx. clu. XXI.. i^cos-S< dx. = x. i.

(68) DIFFERENTIAL CALCULUS. 52. du dx. XXII.. ~l + w. dx. XXIII.. d — cot dx. XXIV.. sec — dx. i. ,. 2. du dx. u. l+u. 2. du. therefore. XX.. 2. —. d vers dx. i. du dx. ##. u-Vu 2. 1 jl. -. 1. du dx. =. •. V2w. y— sin-1 %; sin y = u.. Let. '. By XIIL,. «V« -1. —. XXVI.. Proof of. dx. u. d cosec dx. XXV.. 49.. -1. cos. du dx. 2// =. du — dx. ;. du therefore. But. dy _ dx dx ~ cos y cos y. _. = ± Vl- sin. 2 2/. If the angle y is restricted to the first (Art. 47), cos y is positive.. Hence. and. cosy. dy. dx. = Vl_. du dx. VI. <. = ± VT and fourth quadrants.

(69) ;. .. ;. DIFFERENTIATION Proof of XXI.. 50.. Let y cos y. therefore. — u. du. dx. du dx. dy_ dx~. therefore. But. sin y. If the. = cos _1 w;. dy dx. A, TT ^. angle. ?/. is. 53. smy. = Vl — cos' y = Vl — uK 2. and second quadrants. restricted to the first. (Art. 47), sin y is positive.. Hence. VI -. sin y - =. 51. Proof of XXII.. VI -u. Let y. —. dy dx sec 2 y. But. dy dx. therefore. cot. -1. Proof of XXIII.. u = tan -1 -. = u.. o dy du sec^-^ = dx dx. therefore. 52.. 2. = tan -1 u tan y. therefore. By XV,. ,. du dx. dy_ dx~. and. u2. This. may. _. du dx sec 2 ?/. = 1 -h tai du dx. _ 1. +u. 2. be derived like. XXIL,. or. from.

(70) DIFFERENTIAL CALCULUS. 54 53. Proof sec. u=. -1. cos. XXIV.. of -1. This. may. XXI.. be obtained from. Since. -,. u. =—. -11. cosec. XXV.. Proof of. 54.. =. w. d /1\ dx \uj. d _,1 cos l - = dx u. d _! sec * u dx. mav. du. du. ./. i. I. This. 1. u2 dx. dx. «V% -1 2. i. XX.. be obtained from. Since. - ll sin -> •. u. d. d. _!. , n -1. d 1 f ) dx. 1. —. :. 1. w. du. du dx. u 2 dx. ,-\/ir. ^-i +-i 55.. XXVI.. Proof of. vers _1 w. This. may. XXL. be obtained from. —. Since. = cos -1 (1 — w), d. vers. -1. m. = — cos. x. dx. dx. (1. /-.. _. du. v. — u) Vl-(l-w). V2. 2. EXAM PLES !. 1.. •. 3.. 4.. y ^. 2/. y u. ?/. = 4.-1 tan x. = sec = sin. x. x. -. ———. 5®. 1. -. dy dx. ——. ^.V. 7. (8#. 2. 5.r- — 2a? -h 1. da;. —. = vers -1. o. dt/. 9. ^V4x -9 2. dy. 4. V(aj. _. ),. da;. 5.. 3/. = tan. x. >. 1. _. dx. — 8#. 3. _. %_. - 5)(2 -. 4. VI a. x2. 1. X).

(71) 1. 1. ,. 1. DIFFERENTIATION 6.. 7.. y. ?/. = tan. *. dy d6. (3 tail 0),. = sec -1. 55. dy_. sec 2. 9. .. 10.. 11.. +. r. //. = cosec -1. = tan. *-. x. (-. '. cot. 6x. y. =. cos. 13.. y J. =. o. 14.. >/. -1. Vvers. cot-. ^ +—^a. 16.. y. Sill. = sin _i •. y. A. = sin. 2a e. tan-. dy *c. _. -^. = 0.. *. *=. 6. da;. j/. dy. y. =. sin. -1. y. = cot. ' 1. .,-. 4-a 2)<>2. +&. -. -. , 2. ). a. _. +a. 2 a;. 1.. ty. k -6 2. 1. _ •. +. -a; 2. (. •#_.1. a?). -*. x 4- 1). 2. dx. - cut". 1. (a;. -. e. x. dy 1) da;. 2 '. &a. dy. -. a7.. (a'-ftV 2. ,. (sec x 4- tan. e' 4- e. 19.. 4- sec. dx. da*. 18.. + 2b - 1. SB. f. = tan -1. -2«x. 1. (a;. da;. 17.. e. a V8ar. da;. — COS V2. ;. 2ax_|_. ,. X. + l'. da;. fa;. 15.. 2 *». — = — 4 Vl. a. =. >. sc,. -b. tan- ?. +. +1. Vsec 2. dy dx. 5. 12.. 2. dy _ dx. 1. y=cot -i *"+':' er* — e //. cos 2. 0,. 9. -1 y = vers. -4. o. dO. 8.. 3. _. 2 a;. 4-. 4- e~. x.

(72) DIFFERENTIAL CALCULUS. 56. 20.. y=:tan- l4. + 5tanfl;. 21.. 2/. = cos-. 22.. 2/. =^. 2. members. ^±i = cos-. 23.. 2cos-. 24.. 3 vers- 1 x. 25.. sin. 26.. tan -1 mse +. tan -1. ol-. 27.. -1. ,2. *. a6(l. OQ 29 '. ^. ].. 01 = 2lQg. __. _. z. -\-x. ^. /a;. ,. + tanx). ^. fa \b. ^. .. a;. —x. 2. •. \. ^. _. J. 2. aj. Vl — a ).. 1. -2a; + 5 14 _!^-5 +tan 17^ ^ + 2, + 5 4. 2. 2. = tan -1 ^— 1 — mnxr. "'tan*-* =. tan-. s. dy. &. .^(a^-a —^—- )^ i—. 4. 3 V3 a a. ^. y. 2. 32.. What. /. 2. 2. V4 a - or 2. a?. value must be assigned to a so that the curve. y. may. 2\ x -a. dy = -£. x. .. a-. 54^F+3'. 3. .. 2/=sm. b. 12x 2 -20. =. dy. 2. Q1 31.. sec" 1 -•. a;. a.. 2 +-—=2tan _i \\ —+^2 +3 =[. sin 2a;. of the identical equations, Exs. 23-28.. 1. nse. =2. da;. = vers" [x(2x- 3). a;. » 28.. ^/. + sin -1 a = sin -1 (a y 1 — x. x. vers. 1. +4. % = V6av-a^. =l, --2V^ 2. Differentiate both. 5. da?. ^^-2 x /^. 1. sec- 1. 1. *. ?. 3. =. log e (x. be parallel to the axis of. — 7 a). -f-. tan -1 a«,. X at the point x = 1 ? ^4ns.. J or. — i..

(73) DIFFERENTIATION. 57. A man. walks across the diameter, 200 feet, of a circular courtyard at a uniform rate of 5 feet per second. A lamp at one extremity of a diameter perpendicular to the first casts his shadow upon the circular wall. Required the velocity of the shadow along the wall, when he is at the centre when 20 feet from centre when 33.. ;. 50 feet. ;. when 75. feet. when. ;. ;. at circumference. 10, 9 T8g, 8,. Ans.. 56. Relations between Certain Derivatives.. 6-f,. 5. ft.. per. sec.. It is necessary to notice. the relations between certain derivatives obtained by differentiating. with respect to different quantities.. To. may have. express. — dx. in. terms of. '. —. If y. •. dy. be regarded as a function of -^,. and from the. latter,. dx. is. From. y.. —. a given function of. x,. then x. the former relation,. we. These derivatives are connected. dy. by a simple. relation.. It is evident that. —£ = -—, Ax A# *9. however small the values of Ax and Ay. As these quantities approach zero, we have for the limits of the members of this equation,. w. ft=i dx. That. is,. (i). dx dy. —. the relation between -2 and. -,.. -. dx. ... ordinary tractions.. is. the same as. if. they were. dy a. For example, suppose. w. x=-^~. y Differentiating with respect to. y,. By( i),. •. we have. dx dy. (2). +1. a G/. + 1). 2. !=_M!=-!,. by(2). ..