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:
Then the derivative is
To prove this, we will take the original functions and apply the definition of a derivative.
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.
Rearranging the equation
Factoring out the g(x) from the first two terms and f(x + h) from the last two terms.
Using limit properties, we can reformat the equation.
We can again use the definition of a derivative finds: