Differentiation of a function of a function

It sometimes happens that , instead of being defined directly as
a function of , is given as a function of another variable , which is
defined as a function of . In that case is a function of through
and is called a *function of a function*
or a *composite function*.
The process of substituting one function into another is sometimes
called *composition*.

For example, if , and , then is a function of a function. By eliminating we may express directly as a function of , but in general this is not the best plan when we wish to find .

If and , then is a function of through . Hence, when we let take on an increment , will take on an increment and will also take on a corresponding increment . Keeping this in mind, let us apply the General Rule simultaneously to the two functions and .

- FIRST STEP.
,
.
- SECOND STEP.
- THIRD STEP.
,
.
The left-hand members show one form of the ratio of the increment of each function to the increment of the corresponding variable, and the right-hand members exhibit the same ratios in another form. Before passing to the limit let us form a product of these two ratios, choosing the left-hand forms for this purpose.

This gives , which equals . Write this

- FOURTH STEP. Passing to the limit,

by Theorem 3.8.2.This may also be written

david joyner 2008-11-22