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 .
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
david joyner 2008-11-22