Find the derivative of h(x) = x2ex.
h is a product of the two functions f(x) = x2 and g(x) = ex.
f'(x) = 2x and g'(x) = ex,
While the answer
2xex + 2ex
is correct, we factored out 2ex to find the answer
2ex(x + 1)
because in later problems, it will be helpful to have the answer in this factored form.
While the product rule is usually written
(fg)' = f'g + fg',
the letters "f" and "g" are placeholders. To find the derivative of a product of two functions, find what each function multiplied by the derivative of the other is, and add the results.
If we avoid the letters f and g, the product rule says: to find the derivative of
((first function) × (second function)),
(derivative of first function) × second function
first function × (derivative of second function).
We only need to use the product rule when there are two expressions with x multiplied together.