Product Rule

Definition: Given two differentiable functions, f(x) and g(x), the derivative of f(x)g(x) is f‘(x)g(x)+g‘(x)f(x).

Example 1: Find the derivative of $h(x) = x^2 \sin(x)$.

Solution: This looks like a product of two functions, so let’s break it up. Let f(x) = x^2 and let $g(x) = \sin(x)$; then h(x) = f(x)g(x). We first find the derivatives of these two functions:

f'(x) = 2x; g'(x) = \cos (x)

Then we use the product rule:

h'(x) = f'(x)g(x) + g'(x)f(X) = 2x \sin (x) + \cos (x) x^2

Example 2: Find the derivative of p(x) = 3x \cos (x) e^x.

Solution:  First, we can break p(x) up into three functions: f(x) = 3x); g(x) = \cos (x); h(x) = e^x. Then, find the derivatives of each of these functions:

f'(x) = 3; g'(x) = -\sin (x); h'(x) = e^x

Now, consider p(x) as two functions: f(x)g(x) and h(x). To find the derivative of f(x)g(x), we use the product rule:

Product rule1

Then, we can take the derivative of the entire function:

Product rule2

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