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:

  • h(x) = x ex = (x)(ex)
  • h(x) = x2 sinx = (x2)(sinx)
  • h(x) = e-x2 cos2x = (e-x2)(cos2x)

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 derivatives of f(x) and g(x), the Product Rule provides a formula for the derivative of h(x) = f(x) g(x):
h ′(x) = [f(x)g(x)] ′ = f ′(x) g(x) + f(x) g ′(x),

Proof

We illustrate this rule with the following examples.

  • If h(x) = x ex then
    h ′(x)
    =
    (x) ′ex + x (ex) ′
    =
    ex + xex.
  • If h(x) = x2 sinx then
    h ′(x)
    =
    (x2) ′sinx + (x2)(sinx) ′
    =
    2x sinx + x2 cosx.
  • If h(x) = e-x2 cos2x then
    h ′(x)
    =
    (e-x2) ′cos2x + e-x2 (cos2x) ′
    =
    -2xe-x2 cos2x -2e-x2 sin2x.

Key Concepts [index]

Product Rule

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

h ′(x) = f ′(x)g(x) + f(x)g ′(x).