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:

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:**

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