Chain Rule-Lindsay Wang

Hello class, we are going to learn about the chain rule in today’s class. How exciting, this is one of my favorite topics!

The chain rule is one of the most important concepts in calculus, specifically differentiation. This rule explains how you take the derivative of a "composition of functions", if both are defined in terms of a certain variable.

Suppose you have two functions, f(x) and g(x). The composition of these functions could be:

f(g(x)) or g(f(x))

So what the chain rule says is that the derivative of f(g(x)) = f ' (g(x)) * g'(x), where f'(x) is the derivative of f with respect to x, and g'(x) is the derivative of g with respect to x.

So let's demonstrate this idea in several examples:

Example 1) ln (x^2). This is a composition of two functions, ln x and x^2. 
So the outer function in this case, f(x) is equal to ln x. Thus f'(x) = 1/x.
The inner function in this case, g(x) is x^2, and g'(x) = 2x.

So f(g(x)) has derivative f'(g(x)) * g'(x).

f'(g(x)) = f'(x^2) = 1/x^2
g'(x) = 2x

So the product of f'(g(x)) * g'(x) = 1/x^2 * 2x = 2/x

Example 2) Derivative of (sin x)^2

So here the outer function, f(x) is x^2, which has derivative 2x 
and the inner function, g(x), is sin x, which has derivative cos x

so f(g(x)) has derivative f'(g(x)) * g'(x) = f'(sin x) *cos x = 2sin x cos x

Example 3) Derivative of tan(ln x^2))

This is an example of a composition of more than 2 functions. The three functions in this case are:

f(x) = tan x, so f'(x) = (sec x)^2
g(x) = ln x, g'x) = 1/x 
h(x) = x^2, h'(x) = 2x

Although there are three functions in this case you still apply the same rules: the derivative of the outer function applied to the inner function, times the derivative of the inner function.

So f(g(h(x))) has derivative f'(g(h(x)) * g'(h(x)) * h'(x) =(sec(g(h(x)))^2 * 1 / h(x) * 2x =

(sec(ln x^2))^2 * 1/x^2 * 2x = (sec(ln x^2))^2 * 2/x

These are just the basics about using the chain rule. You just need to find which functions form the composition of the function, find the outer function's derivative, apply it to the inner function, then the inner function's derivative etc etc etc.

Okay class, I hope you enjoyed this lecture as much as I did. Its almost final time, keep up the hard work, you guys will do great!!