# Product Rule

*The Derivative tells us the slope of a function at any point.*

## Product Rule

The product rule tells us how to differentiate the product of two functions:

(fg)’ = fg’ + gf’

Note: the little mark ’ means "Derivative of", and f and g are functions.

### Example: What is the derivative of cos(x)sin(x) ?

The Product Rule says:

(fg)’ = f g’ + f’ g

In our case:

- f = cos
- g = sin

We know (from Derivative Rules):

- cos(x) = −sin(x)
- sin(x) = cos(x)

So:

the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)

= **cos ^{2}(x) − sin^{2}(x)**

## Why Does It Work?

When we multiply two functions f(x) and g(x) the result is the **area fg**:

When we increase x a little, both f and g will change a little also (by Δf and Δg). In this case they both increase making the area bigger.

How much bigger?

Increase in area = fΔg + ΔfΔg + gΔf

As the change in x heads towards zero, the "ΔfΔg" term also heads to zero, and we get:

(fg)’ = fg’ + gf’

## Alternative Notation

An alternative way of writing it (called Leibniz Notation) is:

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

## Three Functions

For three functions multiplied together we get this:

(fgh)’ = f’gh + fg’h + fgh’