The Chain Rule

Sometimes one is stumped by finding that the expression to be differentiated is too complicated to tackle directly.

Thus, the equation is awkward to a beginner.

Now the dodge to turn the difficulty is this: Write some symbol, such as , for the expression ; then the equation becomes which you can easily manage; for Then tackle the expression and differentiate it with respect to , Then all that remains is plain sailing;

for that is, and so the trick is done.

By and bye, when you have learned how to deal with sines, and cosines, and exponentials, you will find the chain rule of increasing usefulness.

Examples

Let us practice using the Chain Rule on a few examples.

Example 9.1. Differentiate .

Solution. Let .

Example 9.2. Differentiate .

Solution. Let .

Example 9.3. Differentiate .

Solution. Let .

Example 9.4. Differentiate .

Solution. Let .

Example 9.5. Differentiate .

Solution. Write this as .

(We may also write and differentiate as a product.)

Proceeding as in example 9.1 above, we get

Hence or

Example 9.6. Differentiate .

Solution. We may write this

Differentiating , as shown in example (2) above, we get so that

Example 9.7. Differentiate .

Solution. Let .

Now let and . Hence

Example 9.8. Differentiate .

Solution. We get

Let and .

Hence or

Example 9.9. Differentiate with respect to .

Solution.

Example 9.10. Find the first and second derivatives of .

Solution.

Let and let ; then .

Hence

Now

(We shall need these two last derivatives later on. See exercise 11 from Chapter 12.)

Example 9.11. A cylinder whose height is twice the radius of the base is increasing in volume, so that all its parts keep always in the same proportion to each other; that is, at any instant, the cylinder is similar to the original cylinder. When the radius of the base is feet, the surface area is increasing at the rate of square inches per second; at what rate is its volume then increasing?¹

Solution.

The volume changes at the rate of cubic inches.

Exercises I

Differentiate the following:

Exercise 9.1. .

Answer

Solution

Let . Then and

Exercise 9.2. .

Answer

Solution

Let . Then , and

Exercise 9.3. .

Answer

Solution

Exercise 9.4. .

Answer

Solution

Let . Then

and

Exercise 9.5. .

Answer

Solution

Using the Quotient Rule

To find where , let , then and

Therefore and

Exercise 9.6. .

Answer

Solution

To find , we need to find and .

where . Then

, where . Then

Now using the Quotient Rule:

Exercise 9.7. .

Answer

Solution

Using the Quotient Rule:

Note that to find , let . Then

Exercise 9.8. Differentiate with respect to .

Answer

Solution

Exercise 9.9. Differentiate .

Answer

Solution

To find , first we need to find the derivative of the numerator. To differentiate of with respect to , we can rewrite it as and apply the Chain Rule. Let . Then: Using the Quotient Rule

Exercise 9.10. A spherical balloon is increasing in volume. If, when its radius is feet, its volume is increasing at the rate of cubic feet per second, at what rate is its surface then increasing?²

Answer

At the rate of square feet per second.

Solution

The volume of the balloon is

and the surface of the balloon is

We know

We want to find .

Differentiate both sides of the following equation with respect to time

Since , we have

Now differentiate both sides of , with respect to time

Substitute in the above equation gives

The process can be extended to three or more derivatives, so that .

Examples

Example 9.12. If ;;, find .

Solution. We have

Example 9.13. If ;;, find .

Solution. Since to calculate , we first need to find , , and .

Differentiating with respect to gives Placing and into the above expression, we get To simplify, multiply both the numerator and denominator by :

So Hence an expression in which must be replaced by its value, and by its value in terms of .

Example 9.14. If ;;and , find .

Solution. We get (see example 9.5); and

So that .

Replace now first , then by its value.

Exercises II

Exercise 9.11. If ;; and , find .

Answer

Solution

Exercise 9.12. If ;; and , find .

Answer

Solution

Exercise 9.13. If ;; and , find .

Answer

Solution

Examples

Exercises I

Examples

Exercises II

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