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11.3 Composition of Functions

11.26   Definition (.) Let be sets and let be functions. The composition of and is the function defined by:

i.e., is the set of all points such that is defined. The rule for is

11.27   Example. If and , then

and

Thus

So in this case . Thus composition is not a commutative operation.

If and , then

and

11.28   Exercise. For each of the functions below, find functions and such that . Then find a formula for .
a)
.
b)
.
c)
.

11.29   Exercise. A Let

Calculate formulas for , , , , , and .

11.30   Entertainment (Composition problem.) From the previous exercise you should be able to find a subset of , and a function such that for all . You should also be able to find a subset of and a function such that for all . Can you find a subset of , and a function such that for all ? One obvious example is the function from the previous example. To make the problem more interesting, also add the condition that for some in .

11.31   Theorem (Chain rule.) Let be real valued functions such that
and . Suppose and , and is differentiable at and is differentiable at . Then is differentiable at , and

Before we prove the theorem we will give a few examples of how it is used:

11.32   Example. Let . Then where

Hence

Let . Then where

Hence

Usually I will not write out all of the details of a calculation like this. I will just write:

Let . Then .

Proof of chain rule: Suppose is differentiable at and is differentiable at . Then

 (11.33)

Since is differentiable at , we know that

Hence the theorem will follow from (11.33), the definition of derivative, and the product rule for limits of functions, if we can show that

Since is differentiable at , it follows from lemma 11.17 that
 (11.34)

Let be a generic sequence in , such that . Then by (11.34), we have
 (11.35)

Since is differentiable at , we have

From this and (11.35) it follows that

Since this holds for a generic sequence in , we have

which is what we wanted to prove. To complete the proof, I should show that is an interior point of . This turns out to be rather tricky, so I will omit the proof.

Remark: Our proof of the chain rule is not valid in all cases, but it is valid in all cases where you are likely to use it. The proof fails in the case where every interval contains a point for which . (You should check the proof to see where this assumption was made.) Constant functions satisfy this condition, but if is constant then is also constant so the chain rule holds trivially in this case. Since the proof in the general case is more technical than illuminating, I am going to omit it. Can you find a non-constant function for which the proof fails?

11.36   Example. If is differentiable at , and , then

Also

i.e.,
 (11.37)

I will use this relation frequently.

11.38   Example (Logarithmic differentiation.) Let
 (11.39)

The derivative of can be found by using the quotient rule and the product rule and the chain rule. I will use a trick here which is frequently useful. I have

Now differentiate both sides of this equation using (11.37) to get

Multiply both sides of the equation by to get

This formula is not valid at points where , because we took logarithms in the calculation. Thus is differentiable at , but our formula for is not defined when .

The process of calculating by first taking the logarithm of the absolute value of and then differentiating the result, is called logarithmic differentiation.

11.40   Exercise. A Let be the function defined in (11.39) Show that is differentiable at , and calculate .

11.41   Exercise. Find derivatives for the functions below. (Assume here that is a function that is differentiable at all points being considered.)

a)
.
b)
.
c)
, where is a rational number.
d)
.
e)
.
f)
.
g)
.
h)
.
i)
.
j)
.

11.42   Exercise. Find derivatives for the functions below. (Assume here that is a function that is differentiable at all points being considered.)
a)
.
b)
.
c)
, where is a rational number.
d)
.
e)
.
f)
.
g)
.
h)
.
i)
.
j)
.

11.43   Exercise. A Calculate the derivatives of the following functions. Simplify your answers.
a)
.
b)
.
c)
.
d)
.
e)
.
f)
.
g)
.
h)
.
i)
.
j)
.
k)
.
l)
.

Next: 12. Extreme Values of Up: 11. Calculation of Derivatives Previous: 11.2 Some General Differentiation   Index
Ray Mayer 2007-09-07