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 and let $g(x) = \sin(x)$; then . We first find the derivatives of these two functions:
Then we use the product rule:
Example 2: Find the derivative of .
Solution: First, we can break p(x) up into three functions: . Then, find the derivatives of each of these functions:
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:
Then, we can take the derivative of the entire function:
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