CHAIN RULE

A frequently used rule in differential calculus is the chain rule. This rule links together derivatives that have related variables. The chain rule is

where the variable y = f(u) is a differentiable function of u and

u = g(x) is a differentiable function of x.

EXAMPLE: Find the derivative of

SOLUTION.- Let

and

Then,

and

Now,

and substituting into equation (1) gives

but,

Therefore,

EXAMPLE: Find where

and

SOLUTION.- By the chain rule

and

Then,

and by substitution

PRACTICE PROBLEMS:

Find in the following:

ANSWERS: