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 f(x) = x^3. The derivative would be simply f'(x) = 3x^2.

But what if was expressed as f'(x) = 3x^2 * 1? Where did this 1 come from? Let’s try this:

f'(x) = 3x^2 \dfrac{d}{dx} x
The 1 came from the derivative of x with respect to x.

So what did we do above? We took the derivative of x^3 and then multiplied it by the derivative of x.

The Chain Rule

Given a (continuous) function h(x) = f(g(x)) where and f(x) and g(x) are different (continuous) functions.

Then \displaystyle h'(x) = f'( g(x) ) g'(x).

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


Example 1:

The function x^3 from earlier has g(x) = xf(x) = x^3f'(x) = 3x^2 and g'(x) = 1. The derivative of x^3 is simply 3x^2.

Example 2:

Consider the function h(x) = \sin(2x). The outside function is f(x) = \sin(x) with f'(x) = \cos(x). The inside function is g(x) = 2x with g'(x) = 2.

By Chain Rule, the derivative h'(x) is 2 \cos(2x).

Example 3 (Combining with Product Rule):

Suppose that we have h(x) = x e^{x^2}. Through product rule and chain rule on the derivative of e^{x^2} gives:

\begin{array}{lcl} h'(x) & = & e^{x^2} + x * e^{x^2} * 2x \ & = & e^{x^2} (1 + 2x^2) \ \end{array}
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 h(x) = \cos((2x + 1)^2). The derivative h'(x) is:

\begin{array}{lcl} h'(x) & = & - \sin(x^2) \dfrac{d}{dx} (2x + 1)^{2} \ & = & - \sin(x^2) * 2(2x + 1) * \dfrac{d}{dx} (2x + 1) \ & = & -2 \sin(x^2) (2x + 1)*2\ & = & -4 \sin(x^2) (2x + 1) \end{array}.

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 x, cos(x), \dfrac{1}{x} and so on.

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