Evaluating Derivatives (Part 2)

In Evaluating Derivatives, we covered the following methods of solving derivatives:

  • Constant Rule
  • Power Rule
  • Coefficient Rule
  • Sum Rule
  • Difference Rule
  • Derivatives you should memorize

In this article, we'll cover the following methods:

  • Product Rule
  • Quotient Rule
  • Chain Rule

Product Rule

Whenever we're taking the product of two functions, we'll utilize the product rule to find the function's derivative!

I know... this looks a little hard to understand at first glance. Essentially, it takes the derivative of the first function and multiplies it by the second function...

...and then adds that to the derivative of the second function multiplied by the first function.

Let's solidify this concept with an example. Say we needed to find the derivative of the following function:

In this case, f(x) as x2...

...and g(x) = sin(x).

Therefore, we can apply the product rule like so:

...which, when we solve out, results in the following derivative function!

Quotient Rule

What if instead of taking the product of two functions, we were dividing (a.k.a. finding the quotient) of them?

We'd use the quotient rule!

This one is a bit more tedious to do, but it’s not as difficult as you’d think.

Imagine that we were given the following function to take the derivative of:

We'd first identify f(x) as x2...

...and then identify g(x) as ln(x).

Then, we'd apply the quotient rule like so, plugging in f(x) = x2 and g(x) = ln(x)...

...and we'd solve like so!

Chain Rule

We covered all of the basic functions in the methods above, but what if we started combining functions into each other and making composite functions?

A composite function is a function made up of another function.

We'd apply the chain rule!

When dealing with a composite function, all we need to do is take the derivative of the outside function f(x) (keeping the function g(x) inside the same)...

...and then multiply by the derivative of the inside function g(x).

It’s not very difficult, but it can be time-consuming if you have several nested functions.

For the sake of solidifying this chain rule application, let's take the derivative of the following function:

We'd identify our inside function g(x) as x2...

...and our outside function f(x) as sin(x2)...

...and apply the chain rule like so...

...which, when we solve out results in the following derivative function!

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