## Differentiation - The Product Rule

You may know hoe to differentiate a simple function such asorGenerally 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 functionsandmultiplied together, then

Example: Differentiate

It is a good habit to get into to write downand 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 functionsandmultiplied together, then

Example: Differentiate
$h=x e^x sin x$

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

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

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

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