Product Rule for Derivatives

In the Calculus and its applications we often encounter functions that are expressed as the product of two other functions, like the following examples:

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In each of these examples, the values of the function h can be written in the form

h(x) = f(x) g(x)

for functions f(x) and g(x). If we know the derivative of f(x) and g(x), the Product Rule provides a formula for the derivative of h(x) = f(x) g(x):

We illustrate this rule with the following examples.

• If then
  
  
  

• If then
  
  
  

• If then
  
  
  

Product Rule

Let f(x) and g(x) be differentiable at x. Then h(x) = f(x) g(x) is differentiable at x and

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