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,

 (5.1)

by Theorem 3.8.2.This may also be written

The above formula is sometimes referred to as the chain rule for differentiation. If and , the derivative of with respect to equals the product of the derivative of with respect to and the derivative of with respect to .

david joyner 2008-11-22