Using the Chain Rule to Differentiate Complex Functions

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  • 0:06 Using the Chain Rule
  • 1:31 Understanding the Chain Rule
  • 2:59 Solving Using the Chain Rule
  • 9:04 Lesson Summary
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Lesson Transcript
Instructor: Erin Monagan

Erin has been writing and editing for several years and has a master's degree in fiction writing.

If you've ever seen a complicated function, this lesson is for you. Most functions that we want to differentiate are complicated functions, for which no single derivative rule will work. In this lesson, learn how to use the chain rule to simplify nesting equations.

Using the Chain Rule

Graph of the parabola for equation f(x) = (2x - 4) squared
Derivative of fx equation

Let's say you're holding a rope that's connected on one end to a wall and the other end is in your hands. This rope may follow the path of a parabola, but let's make it a little bit more realistic. If you've tacked the rope up 16 feet high on one wall, and you're four feet away from that wall holding the other end of the rope, this parabola is given by the equation f(x) = (2x - 4)^2. It's not just x^2 anymore. Now you want to find what the slope is of this parabola at any given point. So, what is the slope close to your hands? What is the slope at the bottom? What is the slope next to the wall? How do you find that?

You need to find the derivative of f(x). What is f`(x)? If f(x)=x^2, then f`(x)=2x. Using your rule of powers, where if f(x)=x^n, then f`(x) = nx^(n - 1). But, what happens in the case of f(x) = (2x - 4)^2? Is the derivative equal to 2x? Is it equal to maybe 4x (since you've got the extra 2 in there)? What is it?

Understanding the Chain Rule

To calculate this and other more complicated derivatives, you need to know the chain rule. The chain rule is used for linking parts of equations together or for differentiating complicated equations like nested equations. So if you have f(x) and this function is really g(h(x)), you've got a function inside of another function. If you have x^2, but instead of it being an x you've got sin x, then you've got one equation, sin x, inside of another equation, x^2.

The chain rule is actually quite simple: Use it whenever you see parentheses. Sometimes, you'll use it when you don't see parentheses but they're implied. But the rule of thumb is that when you see parentheses you're going to use the chain rule. To apply it, take derivatives from the outside in. So if you have f(x) = g(h(x)), then you're going to differentiate the outer function. Then you're going to multiply it by the derivative of the inner function. So for f(x)=g(h(x)), the derivative of f(x) is f`(x) = g`(h(x)) * h`(x).

Solving Using the Chain Rule

Using the chain rule, plug 2x - 4 into the parentheses
Solving using the chain rule

Let's look at what this means in practice. If we have some function - for example, f(x) = (2x - 4)^2 - then we really have two functions. Our first function is the parentheses squared, and our second function is what's inside, 2x - 4.

First, I'm going to ignore the inside for a second and just call it 'parentheses squared'. I'm going to take the derivative of parentheses squared, which would be 2 times whatever is in the parentheses. Then, I need to multiply that by the derivative of whatever is in the parentheses. So if I plug in what's in the parentheses, 2x - 4, I have 2(2x - 4) * d/dx(2x - 4). The derivative is just 2(d/dx)x - (d/dx)4. The derivative of x with respect to x is 1, and the derivative of 4 is zero. I then get 2(2x - 4) * 2 or, simplified, 4(2x - 4).

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