# with sample problems

## Intro

Remember that “product” means the same as multiplication. For example, the product of $3$ and $4$ is $12$, because $3 \cdot 4 = 12$.

Therefore, the Product Rule is used to find the derivative of the multiplication of two or more functions.

## When to use

You know functions are multiplying if they either have a dot, parenthesis or sometimes nothing in between them. Here are some examples to illustrate:

$$f(x)=2x^3 \cdot 5x^{-2}$$

$$f(x)=(2^x + 6x)(5x^7 - 2)$$

$$f(x)= 4x^2 6x^4$$

All three of these functions $f(x)$ are composed of two functions multiplying each other.

• In the first case, the dot in between them means multiplication
• In the second case, they have parenthesis between them, which means they are multiplying
• And in the last case, nothing really separates the functions. In math, when you see two terms (such as $4x^2$ and $6x^4$) with nothing in between them, that means they are multiplying

If there are more than two functions multiplying, you should be able to group these functions to make it so that there are only two functions multiplying.

## How to apply

How easy would it be if the derivative of the product would be equal to the product of each individual derivative? But it's not! Learn below how to actually apply the Product Rule.

## Learn the concept

In words: The derivative of a product of two functions is equal to the derivative of the first times the second, plus the first times the derivative of the second

In math notation: If $f(x)=g(x) \cdot h(x)$, then $f'(x)=g'(x) \cdot h(x) + g(x) \cdot h'(x)$

Let's examine a sample problem below.

## Sample problem

Given that $f(x)=(3x^4+5)(6x^{10})$,

Question 1Find $f'(x)$

Because $f(x)$ is the multiplication of two functions, $(3x^4+5)$ and $(6x^{10})$, we will use the Product Rule to find its derivative.

First, we multiply the derivative of the first by the second (we use the Power Rule, Sum Rule and Constant Rule on the first function),

$$(3x^4+5)'(6x^{10})$$

$$=(12x^3+0)(6x^{10})$$

$$=(12x^3)(6x^{10})$$

Finally, we multiply the coefficients and add the exponents together,

$$=72x^{13}$$

Moving on to the next step, we multiply the first by the derivative of the second (we use the Power Rule for the second function),

$$=(3x^4+5)(6x^{10})'$$

$$=(3x^4+5)(60x^9)$$

We then distribute the derivative of the second (red) into the first (blue),

$$(\color{blue}{3x^4+5})(\color{red}{60x^9})$$

$$=180x^{13}+300x^9$$

Finally, for the last step, we add the results in from steps 1 and 2,

$$f'(x)=72x^{13}+180x^{13}+300x^9$$

Technically, we are done here. But because the first two terms have the same variable and exponent ($x^{13}$), we can simplify those two by adding the coefficients together, for our final answer,

$$f'(x)=252x^{13}+300x^9$$

## Lesson Review

• The Product Rule is used to find the derivative of a multiplication (product) of two functions
• The Product Rule says that the derivative of a product of two functions is equal to the derivative of the first times the second, plus the first times the derivative of the second

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