The Product Rule is an important tool for finding the derivative of a function made up of two other functions multiplied together.

If we have a function like ( h(x) = f(x)g(x) ), we can find its derivative ( h'(x) ) using the Product Rule:

$h'(x) = f'(x)g(x) + f(x)g'(x)$

When to Use the Product Rule

You should use the Product Rule anytime you see two functions being multiplied together.

For example:

• ( x^2\sin(x) )
• ( e^x\ln(x) )

These types of functions are great for this rule.

How the Product Rule Works

To understand the Product Rule, we start with the limit definition of the derivative.

It looks like this:

[ h(x) = f(x)g(x) ]

The derivative is:

[ h'(x) = \lim_{\Delta x \to 0} \frac{h(x + \Delta x) - h(x)}{\Delta x} ]

If we expand ( h(x + \Delta x) ), we have:

[ h(x + \Delta x) = f(x + \Delta x)g(x + \Delta x) ]

Using limits and the rules of derivatives lets us prove our Product Rule formula.

Step-by-Step Examples

1. Example 1: Let ( f(x) = x^2 ) and ( g(x) = e^x ).

   • Here, ( f'(x) = 2x ) and ( g'(x) = e^x ).
   • Using the Product Rule, we find: $h'(x) = 2x e^x + x^2 e^x = e^x(2x + x^2)$

2. Example 2: Let ( f(x) = \cos(x) ) and ( g(x) = \ln(x) ).

   • Then, ( f'(x) = -\sin(x) ) and ( g'(x) = \frac{1}{x} ).
   • Using the Product Rule gives us: $h'(x) = -\sin(x) \ln(x) + \cos(x)\frac{1}{x}$

Practice Problems

Try these problems to practice using the Product Rule:

1. Find the derivative of ( h(x) = (3x^2)(\sin(x)) ).
2. Differentiate ( h(x) = (x^3)(e^{2x}) ).
3. If ( h(x) = (x)(\ln(x)) ), compute ( h'(x) ).

These problems will help you understand when and how to use the Product Rule!