Produt Rule

The product rule allows us to find the derivative of two functions when they are multiplied together. The product rule states that when you have the situation:

Product Rule - 1.png

Then the derivative is

Product Rule - 7.png

To prove this, we will take the original functions and apply the definition of a derivative.

Product Rule - 2.png

The next step is to both add and subtract f(x + h)g(x)/h. We can do this because they will cancel each other out and so we can not changing the equation.

Product Rule - 3.png

Rearranging the equation

Product Rule - 4.png

Factoring out the g(x) from the first two terms and f(x + h) from the last two terms.

Product Rule - 5.png

Using limit properties, we can reformat the equation.Product Rule - 6.png

We can again use the definition of a derivative finds:

 

Product Rule - 7