The Chain Rule Examples

Let

Without rewriting f, identify the inside and outside functions.

If we were to evaluate this function on a calculator, first we would need to find x², therefore x² is the inside function. x² provides the input to the function

therefore  is the outside function.

Find the derivative of h(x) = cos(sin x).

We can think of this function as f(g(x)), where

f(□) = cos(□)

and

g(x) = sin x.

Then

f ' (□) = -sin (□),

therefore

f ' (g(x)) = -sin(sin x).

Multiplying by

g ' (x) = cos x,

we find

h ' (x) = f ' (g(x)) · g'(x) = -sin(sin x) · cos x.

Find the derivative of h(x) = sin (x²).

• First we identify the outside and inside functions. The outside function is the one on the outside:

sin(□)

and the inside function is the one on the inside:

x².

• The derivative of the outside function is

cos (□).

Using the inside function (x²) as the input, we find

cos(x²).

• Finally, we multiply by the derivative of the inside function. The derivative of the inside function is

(x²)' = 2x,

so the final answer to the problem is

h ' (x) = cos(x²) · 2x.

Find the derivative of the function h(x) = (ln x)².

The outside function is

f(□) = (□)²

and the inside function is

g(x) = ln x.

The derivative of the outside function is

f ' (□) = 2(□),

and if we use the inside function as the input we find

f ' (g(x)) = 2(ln x).

The derivative of the inside function is

multiplying the appropriate things together we find our final answer: