The Chain Rule (Simple)

Calculus Tutorial

The Chain Rule (Simple)


Here, I will be talking about how to find the derivative of a function using The Chain Rule.

To start off, I’ll remind you, or tell you, what the derivative is.

The derivative of a function is the rate of change of that function at the given point. This is also known as the slope of the tangent line at the given point on a graph. If any of that is confusing to you, just think: how fast the function is changing at a point. I’ll go over this again as we do the problem.

Sample Problem

Find the derivative of f(x) = 2(x^3)^4 using the chain rule.


Find the derivative of f(x) = 2(x^3)^4 using the chain rule.

Step 1.
f(x) = 8(x^3)^3

In this step, we are multiplying the coefficient of the function by the fourth power that is on the outside. We then reduce that fourth power into a cubic power (a power of 3).

Step 2.
f(x) = 8(x^3)^3 * 2x^2

Here, we are essentially doing the same thing again except this time we are multiplying the function by what we get which is to say that our goal is to find the derivative of the inside (inside the parentheses) and multiply the whole function by that. We’re pulling the derivative of the inside out with a “chain”.

Step 3.
f'(x) = 8(x^3)^3 * 2x^2

This is the final answer for our question. As a few final guidance points, I’d like to tell you how I always remembered the chain rule: “Derivative of the outside multiplied by the derivative of the inside” or “You have to be outside before you can go inside”. The first one is a simplistic, but correct view of the chain rule process and the latter of the two is a way to help you remember which goes first without using mathematical terms.

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