Basic Rules of Differentiation (Part 3)

In Part 1 and Part 2, we learned the rules of differentiation to handle common functions we might encounter. Now, we will learn about chain rule which will close this chapter of the basic rules of differentiation.

Chain Rule:

Chain rule is perhaps the most powerful of all rules of differentiation. This is due to the fact that many other rules of differentiation are based off it. In order to understand this rule, consider a function f(x) that is a composite function of the functions g(x) and h(x). In this case, the function h(x) is inside the function g(x).

It can be observed that g(x) is the outside function, and h(x) is the inside function. Then, the derivative of f(x) is given by:

This is known as chain rule. It basically says the derivative of f(x) is "the derivative of the outside (keep the inside function the same) times the derivative of the inside."

Examples:

Let's observe some examples of how we can apply chain rule. It is important to understand that the other rules of differentiation still apply.

Once we know how to apply chain rule, we can say that we know how to differentiate. Chain rule is the "pillar" of differentiation. If we understand it, we will be able to learn more complex rules of differentiation and even integration.