# Chain Rule

This post will be about the chain rule. The chain rule was one of those topics that took a bit of time for me to understand when I was a younger math student. It is assumed that the reader knows about the product rule.

A Motivating Example

Consider a simple function such as . The derivative would be simply .

But what if was expressed as ? Where did this 1 come from? Let’s try this:

The 1 came from the derivative of x with respect to x.

So what did we do above? We took the derivative of and then multiplied it by the derivative of .

The Chain Rule

Given a (continuous) function where and and are different (continuous) functions.

Then .

This means we take the derivative of the outside function and then take the derivative of the inside function . It can be possible that the function inside can be a different function such as which is different from .

Examples

Example 1:

The function from earlier has and . The derivative of  is simply .

Example 2:

Consider the function . The outside function is with . The inside function is with .

By Chain Rule, the derivative is .

Example 3 (Combining with Product Rule):

Suppose that we have . Through product rule and chain rule on the derivative of gives:

Example 4 (Multiple Chain Rule):

There are cases when you may have to use multiple chain rules along with product rules, quotient rules and so on.

Consider . The derivative is:

.

Tips for Learning Chain Rule:

1. Take it one step at a time.
2. Identify the outside function(s) and inside function(s). Start from the outside to the inside.
3. Practice with the simple functions such as , , and so on.