The natural exponential function has an inverse that “undoes” its work. This inverse is known as the natural logarithm, or simply the natural log.
What’s a natural log?
Alogarithm of a quantity is a power to which a positive real constant is raised to get that quan-tity. The constant is called the base, which is usually 10 or e. We’ll deal only with the base-e or natural log.
In equations, the natural log is usually denoted by writing “ln” followed by the argument.
Here are some equations that you can check out with your calculator. With the exceptions of
lne, ln 1, and ln (1/e), which are exact values, everything has been rounded to three decimal
While the natural log “undoes” the natural exponential, the reverse operation also works.
Therefore,
ln (ex ) =x and
e (ln x )=x
Thenatural log function is defined only for positive real-number arguments. In the first equation above, that’s not a problem, because e x is always a positive real, no matter what real number we input for x. In the second equation, we must restrict the domain to the positive reals only. As long as we confine the domain to the positive reals, the natural log function is continuous and differentiable.
What does it look like?
Figure 7-6 is a graph of the natural log function. Its domain is the set of positive real numbers, and it’s continuous at every point in that domain. The function is singular (meaning that it blows up) as x approaches 0 from the right, but this is not a problem as long as we keep 0 and all the negative reals out of the domain. Once we restrict the domain in that way, the natural log function doesn’t have any gaps or turn any corners, so it’s differentiable.
At any point on the graph in Fig. 7-6, the slope is equal to 1/x. We can write this as the equation
d /dx (ln x ) = 1/x
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Example
Let’s work out a derivative involving a natural logarithm. Without getting complicated, but still requiring some thought, we’ll tackle this:
p (x ) = ln (7x− 14)
The chain rule will work here. If we call the component functions f and g, and if we express the above function as g ( f ), then
f (x ) = 7x− 14 and
g ( y ) = ln y
The function f is differentiable over the entire set of reals, but g is differentiable only as long asy > 0. That means we must restrict the domain of g ( f ) so that
7x− 14 > 0
f(x) = ln x f′ (x) = 1/x
2 4 6
–2 –4 –6
2 6
–2
–4
–6
x 4
Value of function
Figure 7-6 The derivative of the natural log function is the reciprocal function.
The domain must be restricted to the set of positive real numbers.
This works out to x > 2. Now we can differentiate both functions to get f ′(x ) = 7
and
g ′( y ) = 1/y
The chain rule, stated once again, tells us that if f and g are differentiable functions of the variable x, then
{g [ f (x )]}′ =g ′ [ f (x )] · f ′(x ) Substituting the values into this formula, we get
p′(x ) = [1 / (7x− 14)] · 7 = 7 / (7x− 14)
We can multiply both the numerator and denominator by 1/7, reducing this to p′(x ) = 1 / (x− 2)
Some people would rather write this as
p′(x ) = (x− 2)−1
Here’s a challenge!
Suppose we’re confronted with a differentiation problem that involves the composite of a trig function and a log function. Let’s find the derivative of
q (x ) = ln (cos 2x )
We’ll also define the values of x, if any, for which this function is nondifferentiable.
Solution
Before we get started with the differentiation, we had better find out where q is differentiable and where it is not. The natural log function is defined only for positive values of the argument. That means q is dif-ferentiable only for those values of x such that
cos 2x > 0
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This inequality is satisfied for infinitely many open intervals. It’s awkward to express this symbolically, but the following arrangement gives the general idea:
From “negative infinity”
Our function q is defined and continuous everywhere in each of these open intervals. There are no gaps, no singularities, and no corners within any single interval. But we must take note of the fact that these are open intervals, meaning that the end points are not included. Our function q is differentiable at any point within any of these open intervals, but nowhere else. You might find it helpful to graph the function
y= cos 2x
in the xy-plane to see where this function is positive, where it’s zero, and where it’s negative. Then you’ll be able to envision the above open intervals along the x axis.
In this situation, we’re dealing with a function of a function of a function! Another way to say this is to call q a triplet of nested functions. To differentiate it, we must employ the chain rule twice. Let’s call the component functions f, g, and h, working from the inside out. Then we have
q (x ) =h {g [ f (x )]} Applied to the outer two functions, the chain rule gives us
[h ( g)]′ =h ′( g ) · g ′
If we let g operate on f, we can rewrite this as
{h [ g ( f )]}′ =h ′ [ g ( f )] · [ g ( f )]′ The chain rule, applied to the inner pair of functions, tells us that
[g ( f )]′ =g ′( f ) · f ′ So, by substitution, we obtain
{h [ g ( f )]}′ =h ′ [ g ( f )] · g ′( f ) · f ′
The left-hand side of the above equation happens to be the same as q ′, which is what we seek! So let’s substitute:
q ′ =h′ [ g ( f )] · g ′( f ) · f ′
We can include the variable x in the above expression to see how it fits in:
q′(x ) =h ′ {g [ f (x )]} · g ′ [ f (x )] · f ′(x )
Now let’s plug in expressions for what each of these functions does to its argument, and also plug in the arguments themselves. When we do that, we get
q ′(x ) = [1 / (cos 2x )] · (−sin 2x ) · 2 = −2 (sin 2x ) / (cos 2x )
It’s time for us to invoke another well-known law from trigonometry: The sine divided by the cosine is equal to the tangent, abbreviated tan. Knowing this, we can rewrite the above equation as
q ′(x ) = −2 tan 2x
provided 2x is not an odd-integer multiple of p/2. That means x can’t be an odd-integer multiple of p/4.
We’ve already taken this constraint into account. Remember that the original function q is differentiable only within certain open intervals, none of which contains an odd-integer multiple of p/4.
Are you confused?
Way back in the 1960s, one of my math teachers asked the class at the end of an especially difficult session,
“Are you confused by this?” When we all nodded, he said, “I don’t blame you.” The chain rule is tricky enough when applied once. When you apply it twice to a triplet of nested functions, it’s worse! If you’re disoriented by the process we just went through, put it aside for now, and look at it again tomorrow.
Where to find more derivatives
You can find some worked-out derivatives in the back of this book. Refer to App. F. You can also find them on the Internet. Enter the phrase “table of derivatives” into your favorite search engine. A few sites will calculate derivatives for you.
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Real powers in general
A general real-number exponent can be evaluated with natural logs and exponentials. From algebra and precalculus, remember that when you have an expression of the form x k where x is a nonzero variable and k is any real number, then
ln (x k ) =k ln x
Because the natural exponential function “undoes” the natural log function, you can take the natural exponential of both sides of the above equation to get
x k=e(k ln x )
If you want to raise a variable to any real power, you can take the natural log of the vari-able, multiply by the exponent, and finally take the natural exponential of that product. This scheme only works when x > 0, however, because the natural log of 0 or a negative quantity is not defined.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not represent the only way a problem can be figured out. If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it!
1. Differentiate the function
p (t)= 2t2− 4t+ 5 + 4t−1+ 6t−2
Indicate the values of t, if any, for which the derivative is not defined.
2. Differentiate the function
q (w)= (w−1+ 1)(w−1− 1)
Indicate the values of w, if any, for which the derivative is not defined.
3. The reciprocal of the sine function is known as the cosecant function, abbreviated csc.
Mathematically,
cscx= 1 / sin x
provided x is not an integer multiple of p. Using the rules we’ve learned so far, differentiate the cosecant function. Indicate the values of x, if any, for which this function is nondifferentiable.
4. The reciprocal of the cosine function is known as the secant function, abbreviated sec.
Mathematically,
secx= 1 / cos x
provided x is not an odd-integer multiple of p/2. Using the rules we’ve learned so far, differentiate the secant function. Indicate the values of x, if any, for which this function is nondifferentiable.
5. As mentioned in the chapter text, the sine function divided by the cosine function is the tangent function. Mathematically,
tanx= sin x / cos x
provided x is not an odd-integer multiple of p/2. Using the rules we’ve learned so far, differentiate the tangent function. Indicate the values of x, if any, for which this function is nondifferentiable.
6. The cosine function divided by the sine function is known as the cotangent function, abbreviated cot. Mathematically,
cotx= cos x / sin x
provided x is not an integer multiple of p. Using the rules we’ve learned so far, differentiate the cotangent function. Indicate the values of x, if any, for which this function is nondifferentiable.
7. Differentiate this generalized function, where a is a real-number constant:
p (x ) =eax
8. Differentiate this generalized function, where a and b are real-number constants:
q (x ) =beax
9. Differentiate this generalized function, where a is a real-number constant:
r (x ) = ln ax
10. Differentiate this generalized function, where a and b are real-number constants:
s (x ) =b ln ax
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