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

PROCEDURE

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 

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

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