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Differentiation II

The product rule

The product rule

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Explainer Video

Tutor: Toby

Summary

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)f(x)=u(x)v(x)f(x)=u(x)v(x) where u(x)u(x)u(x) and v(x)v(x)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)=(x2−4)sin⁡(x)f(x)=(x^2-4)\sin(x)f(x)=(x2−4)sin(x)


One function is u(x)=x2−4‾\underline{u(x)=x^2-4}u(x)=x2−4​ and the other is v(x)=sin⁡(x)‾\underline{v(x)=\sin(x)}v(x)=sin(x)​.


The product rule

Function f(x)f(x)f(x)​

Derivative f′(x)f'\left(x\right)f′(x)

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

u′(x)v(x)+u(x)v′(x)u'(x) v(x)+u(x) v'(x)u′(x)v(x)+u(x)v′(x)


PROCEDURE

1.

Identify the individual functions u(x)u(x)u(x) and v(x)v(x)v(x) that make up the product.

2.

Find the derivatives of each of these functions: u′(x)u'(x)u′(x) and v′(x)v'(x)v′(x).

3.

The derivative of the product is u′(x)v(x)+u(x)v′(x)u'(x) v(x)+u(x) v'(x)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 xxx.

f(x)=(x3−1)(2x5+1)f\left(x\right)=(x^3-1)(2x^5+1)f(x)=(x3−1)(2x5+1)

Identify the individual functions:

u(x)=x3−1u(x)=x^3-1u(x)=x3−1 and v(x)=2x5+1v(x)=2x^5+1v(x)=2x5+1​​


Differentiate each of these with respect to xxx:

u′(x)=3x2u'(x)=3x^2u′(x)=3x2 and v′(x)=10x4v'(x)=10x^4v′(x)=10x4​​


Compile the derivative by the rule:

f′(x)=u′(x)v(x)+u(x)v′(x) f′(x)=3x2(2x5+1)+(x3−1)10x4f^\prime(x)=u'(x)v(x)+u(x)v'(x)\\\space\\f'(x)=3x^2(2x^5+1)+(x^3-1)10x^4f′(x)=u′(x)v(x)+u(x)v′(x) f′(x)=3x2(2x5+1)+(x3−1)10x4


This can be summed up and simplified:

f′(x)=16x7−10x4+3x2‾\underline{f'(x)=16x^7-10x^4+3x^2}f′(x)=16x7−10x4+3x2​



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)f(x)=u(x)v(x)w(x)f(x)=u(x)v(x)w(x), the first derivative is 


f′(x)=u′(x)v(x)w(x)+u(x)v′(x)w(x)+u(x)v(x)w′(x)\boxed{f'(x)=u'(x)v(x)w(x)+u(x)v'(x)w(x)+u(x)v(x)w'(x)}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)=(x3+2x)(2x2−3x+5)sin⁡(x)f(x)=(x^3+2x)(2x^2-3x+5)\sin(x)f(x)=(x3+2x)(2x2−3x+5)sin(x) with respect to xxx. 


The individual functions that make up the product are u(x)=x3+2xu(x)=x^3+2xu(x)=x3+2x, v(x)=2x2−3x+5v(x)=2x^2-3x+5v(x)=2x2−3x+5 and w(x)=sin⁡(x)w(x)=\sin(x)w(x)=sin(x). Differentiate each of these with respect to xxx:

u′(x)=3x2+2u'(x)=3x^2+2u′(x)=3x2+2, v′(x)=4x−3v'(x)=4x-3v′(x)=4x−3 and w′(x)=cos⁡(x)w'(x)=\cos(x)w′(x)=cos(x)​


The product rule states that for f(x)=u(x)v(x)w(x)f(x)=u(x)v(x)w(x)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)f'(x)=u'(x)v(x)w(x)+u(x)v'(x)w(x)+u(x)v(x)w'(x)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)=(3x2+2)(2x2−3x+5)sin⁡(x)+(x3+2x)(4x−3)sin⁡(x)+(x3+2x)(2x2−3x+5)cos⁡(x)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)f′(x)=(3x2+2)(2x2−3x+5)sin(x)+(x3+2x)(4x−3)sin(x)+(x3+2x)(2x2−3x+5)cos(x)​​


This is simplified to 

f′(x)=(10x4−12x3+27x2−12x+10)sin⁡(x)+(2x5−3x4+9x3−6x2+10x)cos⁡(x)‾\underline{f'(x)=(10x^4-12x^3+27x^2-12x+10)\sin(x)+(2x^5-3x^4+9x^3-6x^2+10x)\cos(x)}f′(x)=(10x4−12x3+27x2−12x+10)sin(x)+(2x5−3x4+9x3−6x2+10x)cos(x)​


​

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FAQs - Frequently Asked Questions

How does the product rule work for functions that are products of more than two functions?

No matter how many functions are in the product, the product rule works in the same way. For the function 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).

What is the product rule?

For a function f(x)=u(x)v(x), the product rule says the derivative is f'(x)=u'(x)v(x)+u(x)v'(x).

What is a product of functions?

A product of two functions is of the form f(x)=u(x)v(x) where u(x) and v(x) are functions themselves.

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