The product rule

In a nutshell

The product rule is used to differentiate functions that are products of functions. For example $f(x)=u(x)v(x)$ where $u(x)$ and $v(x)$ are themselves functions. It also works for functions that are products of more than two functions.

Example 1

Identify the two functions that make up the product of functions

$f(x)=(x^2-4)\sin(x)$

One function is $\underline{u(x)=x^2-4}$ and the other is $\underline{v(x)=\sin(x)}$ .

The product rule

Function $f(x)$

Derivative $f'\left(x\right)$

$u\left(x\right) v\left(x\right)$

$u'(x) v(x)+u(x) v'(x)$

PROCEDURE

1.	Identify the individual functions $u(x)$ and $v(x)$ that make up the product.
2.	Find the derivatives of each of these functions: $u'(x)$ and $v'(x)$ .
3.	The derivative of the product is $u'(x) v(x)+u(x) v'(x)$ . In other words, the sum of the two products of one function multiplied by the derivative of the other.

Example 2

Differentiate the following function with respect to $x$ .

$f\left(x\right)=(x^3-1)(2x^5+1)$

Identify the individual functions:

$u(x)=x^3-1$ and $v(x)=2x^5+1$ 

Differentiate each of these with respect to $x$ :

$u'(x)=3x^2$ and $v'(x)=10x^4$ 

Compile the derivative by the rule:

$f^\prime(x)=u'(x)v(x)+u(x)v'(x)\\\space\\f'(x)=3x^2(2x^5+1)+(x^3-1)10x^4$

This can be summed up and simplified:

$\underline{f'(x)=16x^7-10x^4+3x^2}$

More than two functions

You can also use the product rule when the function is a product of more than two functions. The rule is the same: the derivative is the sum of the products made up of the derivative of one function and the other non-differentiated functions. For example for $f(x)=u(x)v(x)w(x)$ , the first derivative is

$\boxed{f'(x)=u'(x)v(x)w(x)+u(x)v'(x)w(x)+u(x)v(x)w'(x)}$

Example 3

Differentiate the function $f(x)=(x^3+2x)(2x^2-3x+5)\sin(x)$ with respect to $x$ .

The individual functions that make up the product are $u(x)=x^3+2x$ , $v(x)=2x^2-3x+5$ and $w(x)=\sin(x)$ . Differentiate each of these with respect to $x$ :

$u'(x)=3x^2+2$ , $v'(x)=4x-3$ and $w'(x)=\cos(x)$ 

The product rule states that for $f(x)=u(x)v(x)w(x)$ , the derivative is $f'(x)=u'(x)v(x)w(x)+u(x)v'(x)w(x)+u(x)v(x)w'(x)$ . Thus here you have:

$f'(x)=(3x^2+2)(2x^2-3x+5)\sin(x)+(x^3+2x)(4x-3)\sin(x)+(x^3+2x)(2x^2-3x+5)\cos(x)$ 

This is simplified to

$\underline{f'(x)=(10x^4-12x^3+27x^2-12x+10)\sin(x)+(2x^5-3x^4+9x^3-6x^2+10x)\cos(x)}$