FEUILLE DE CALCUL RAPIDE 7 janvier 2019

FEUILLE DE CALCUL RAPIDE 7 janvier 2019

1.

(Ederiv32.tex)

ln x ² + ln x + 1 x 2.

(Ederiv23.tex)

h ⁰ _y (x) = (ln y)y ^x 3.

(Ederiv61.tex)

− 1 x √

1 + x ² 4.

(Ederiv34.tex)

2x (1 + x ³ )

⁵³

5.

(Ederiv48.tex)

− 2 1 + x ² 6.

(Ederiv57.tex)

− cos x 7.

(Ederiv44.tex)

1 sin x 8.

(Ederiv63.tex)

1 sin x

9.

(Ederiv47.tex)

Les fonctions sont d´ erivables dans ]0, 1[. Les d´ eriv´ ees sont

− 1 2 p

x(1 − x) et 0 10.

(Ederiv21.tex)

f (x)f ⁰⁰ (x) f ⁰ (x) ² 11.

(Eexo12.tex)

1 − th ² x 12.

(Eexo107.tex)

x − 1 ln x 13.

(Ederiv45.tex)

]1, +∞[ 1 + ln(ln x) x 14.

(Ederiv53.tex)

4x 1 + x ⁴ 15.

(Ederiv62.tex)

t 7→ x cos t cos(x sin(t)).

16.

(Ederiv11.tex)

1 2 √

k + 1 ≤ √

k + 1 − √ k ≤ 1

2 √ k 17.

(Ederiv29.tex)

− 1 cos x 18.

(Ederiv18.tex)

− 10x ² + 10x + 1 (x ² + x + 1) ⁴ 19.

(Eexo66.tex)

(−1) ⁿ⁻¹ (n − 1)!

(1 + x) ⁿ 20.

(Ederiv4.tex)

−2 sin t

21.

(Ederiv6.tex)

−2 x ² − 1 22.

(Ederiv60.tex)

cosec ⁰ (x) = −cosec(x) p

(cosec(x)) ² − 1

arccosec ⁰ (y) = − 1 y p

y ² − 1 23.

(Ederiv55.tex)

1 cos ³ x 24.

(Ederiv15.tex)

1 cos x − 1 25.

(Ederiv39.tex)

2

(cos θ − sin θ) ² = 2 1 + tan ² θ (1 − tan θ) ² 26.

(Ederiv24.tex)

la fonction nulle

27.

(Ederiv64.tex)

ϕ ⁰ (−1) = ln(n!) n!

28.

(Ederiv16.tex)

1 1 + cos x 29.

(Ederiv7.tex)

1 + th t 1 − th t

0

= 2 1 + th t 1 − th t Cela traduit que

1 + th t 1 − th t = e ^2t . 30.

(Eexo67.tex)

(x ² + (2n + 1)x + 1 + n ² )e ^x 31.

(Eexo180.tex)

−f (−x)

32.

(Ederiv54.tex)

e ^x arctan(e ^x ) 33.

(Ederiv65.tex)

− 2x 1 + x ⁴ 34.

(Ederiv9.tex)

√ 1 x ² + 1 35.

(Ederiv51.tex)

2 1 + x ² 36.

(Ederiv46.tex)

f _k ⁰ = kf _k−1 + (k + 1)f _k+1 37.

(Ederiv41.tex)

− tan x

38.

(Ederiv3.tex)

2(ln x + 1)x ^2x 39.

(Ederiv12.tex)

oui ´ evidemment ! ! 40.

(Ederiv19.tex)

1 f ⁰⁰ ◦ ϕ

1 ACDeriv

FEUILLE DE CALCUL RAPIDE 7 janvier 2019

41.

(Ederiv37.tex)

1 1 − sin x 42.

(Ederiv14.tex)

Faux.

43.

(Ederiv49.tex)

1 1 + x ² 44.

(Ederiv66.tex)

1 + i tan t 1 − i tan t

0

= 2i

1 + i tan t 1 − i tan t

On retrouve ce r´ esultat en v´ erifiant facilement que 1 + i tan t

1 − i tan t = e ^2it 45.

(Ederiv25.tex)

2 1 + x ² 46.

(Ederiv38.tex)

arctan x

47.

(Ederiv35.tex)

1 ch 48.

(Eexo126.tex)

(ln 2) 2 ^x + (ln 3) 3 ^x 49.

(Ederiv52.tex)

p x ² + λ 50.

(Eexo182.tex)

(−1) ⁿ n!

x ⁿ⁺¹ 51.

(Ederiv58.tex)

1 cos x 52.

(Ederiv43.tex)

1

sin x

53.

(Ederiv5.tex)

arctan x 54.

(Ederiv8.tex)

(1 − ln x)x

^x1

⁻² 55.

(Ederiv2.tex)

(2 ln x + 1)x ^1+x

²

56.

(Ederiv30.tex)

1 (2x ² + 1) ² √

1 + x ² 57.

(Ederiv28.tex)

1 cos x 58.

(Ederiv1.tex)

2x(ln 2)2 ^(x

²

⁾ 59.

(Ederiv56.tex)

1 2 √

x arcsin( √ x)

60.

(Eexo119.tex)

t ³ + 3nt ² + 3n(n − 1)t + n(n − 1)(n − 2) e ^t

61.

(Eexo179.tex)

−2x ² + 1

(4x ² + 2x + 1)(x ² + x + 1) 62.

(Ederiv59.tex)

sec ⁰ (x) = sec(x) p

(sec(x)) ² − 1

arcsec ⁰ (y) = 1 y p

y ² − 1 63.

(Ederiv22.tex)

k ⁰ _x (y) = xy ^x−1

64.

(Ederiv13.tex)

vrai 65.

(Ederiv50.tex)

f (x) + Z x

0

e ^x−t f (t) dt 66.

(Ederiv33.tex)

1

x ln(x) ln(ln(x)) 67.

(Eexo77.tex)

n

X

k=0

n k

f ^(k) g ^(n−k)

68.

(Ederiv42.tex)

1 cos x

69.

(Ederiv20.tex)

ϕ 70.

(Ederiv31.tex)

2x ln x + x 71.

(Ederiv40.tex)

(n ² + 3n + 4)e 72.

(Ederiv17.tex)

ab 1 + tan ² (bx) 1 + a ² tan ² (bx) 73.

(Ederiv26.tex)

2 1 + x ² 74.

(Eexo181.tex)

(ln x + 1)x ^x

75.

(Ederiv36.tex)

1 ch x 76.

(Ederiv10.tex)

√ 1 x ² − 1 77.

(Ederiv27.tex)

− 2 1 + x ²

2 ACDeriv