## Differentiation - The Product Rule

You may know hoe to differentiate a simple function such as or Generally functions are built out of these simple functions to make more complicated functions and we must learn to differentiate these more complicated functions too. The simplest way two functions can be combined to make a more complicated function is to multiply them. Then they can be differentiated using the product rule:

The Product Rule:

If a function h consists of two simpler functions and multiplied together, then Example: Differentiate It is a good habit to get into to write down and then you can just substitute them into the expression for   Example: Differentiate   The product rule can be used repeatedly with any number of products.

If a function h consists of three simpler functions and multiplied together, then Example: Differentiate
$h=x e^x sin x$

$e=x, \: f=e^x , \: g = sin x$

$\frac{de}{dx} =1, \: \frac{df}{dx} =e^x, \: \frac{dg}{dx} = cos c$

$\frac{dh}{dx} =x e^x coxs x + xe^x sin x + e^x sin x$ 