## Product Rule Formula

We use the product rule when we have 2 functions multiplied together. Don’t be on the lookout for a multiplication sign because there often won’t be one. Instead look for any instances of two different functions of x next to each other

The formula for the product rule is provided in the formula sheet:

$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$

When you are looking at the product rule, all you are doing is breaking a complicated function down to simpler functions, that we can easily differentiate.

## Product Rule Differentiation Example

Let’s look at 2014 exam 1 Q1a

For $y=x^{2}sin(x)$

Find  $\frac{dy}{dx}$

We need to separate this function into two functions and set one as u and one as v, it doesn’t effect the final result which we set to u and which we set to v.

u = $x^{2}$

v = $sin(x)$

It is important to define the terms before we start using them. If you didn’t spell out what u is equal to, the examiner doesn’t know what u is and where it came from.

Using our basic power and trigonometric differentiation rules we can work out that:

$\frac{du}{dx}=2\newline \newline \frac{dv}{dx}=cos(x)$

All we have to do now is substitute these values into our original formula:

$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\newline =x^{2} \times cos(x)+ sin(x) \times 2 \newline =x^{2}cos(x)+ 2sin(x)$