## Chain Rule.

**Definition:**

The chain rule is a method for differentiating a function of a function, or differentiating composite functions.

Consider the expression . We notice that this is not a normal sine function. It has an as argument for the sine function. Therefore, we can consider the in the sine function as a whole different function. This can be broken into two functions, and .

If we consider , we can write .

In order to differentiate a composite function, , i.e; to find , we let

and

What this implies is that whatever is, **u** will be equal to that. Then, the process of differentiating is to find

- (as
**u**will be a function of**x**).

Finally, we can write,

Back to our example, ; remember that and . We let

and

Therefore, and

This leaves us with . We can simplify the equation by writing

Note that the **u** in the cosine function is replaced with .…