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