At a Glance - Derivative of a Product of Functions
Use the product rule whenever there is a function that's a product of two other functions of x. If
f(x) = ({something with an x}) × ({something else with an x}),
then use the product rule to find f'.
To use the product rule, figure out which two functions are multiplied together, and the derivative of each of those functions.
Example 1
Find the derivative of h(x) = x^{2}e^{x}. |
Example 2
Find the derivative of f(x) = sin x cos x. |
Example 3
Find the derivative of g(x) = x^{2}sin x |
Example 4
Find the derivative of f(x) = 4x^{2}. |
Example 5
Find the derivative of f(x) = x^{2}e^{x}sin x. |
Exercise 1
Let f(x) = x + 1 and g(x) = x.
- Find f'(x).
Exercise 2
Let f(x) = x + 1 and g(x) = x.
- Find g'(x).
Exercise 3
Let f(x) = x + 1 and g(x) = x.
- Find f'(x) × g'(x).
Exercise 4
Let f(x) = x + 1 and g(x) = x.
- Find (f × g)'(x).
Exercise 5
Let f(x) = x + 1 and g(x) = x.
- Must the derivative of the function f × g be equal to the product of f' and g'?
Exercise 6
Find the derivative of each function. They may not all require the product rule.
- f(x) = xsin x
Exercise 7
Find the derivative of each function. They may not all require the product rule.
- f(x) = e^{x}cos x
Exercise 8
Find the derivative of each function. They may not all require the product rule.
- f(x) = xln x
Exercise 9
Find the derivative of each function. They may not all require the product rule.
- g(x) = 5^{x}e^{x}
Exercise 10
Find the derivative of each function. They may not all require the product rule.
- g(x) = (log_{2 }x)(log_{3 }x)
Exercise 11
Find the derivative of each function. They may not all require the product rule.
- g(x) = 5e^{x}
Exercise 12
Find the derivative of each function. They may not all require the product rule.
- h(x) = (x^{2} + 2x)ln x
Exercise 13
Find the derivative of each function. They may not all require the product rule.
- h(x) = ln x cos x
Exercise 14
Find the derivative of each function. They may not all require the product rule.
Exercise 15
Find the derivative of each function. They may not all require the product rule.
- j(x) = ln x + cos x
Exercise 16
Find the derivative of f(x) = (x^{2} + 2)(x^{3}-4).
- Use the product rule.
Exercise 17
Find the derivative of f(x) = (x^{2} + 2)(x^{3}-4).
- Rewrite f by multiplying the factors together, then take the derivative.
Exercise 18
- Find the derivative of
f(x) = x^{2}e^{x}sin x
thinking of the function as
f(x) = (x^{2})(e^{x}sin x).
Exercise 19
Find the derivative of the function
f(x) = x^{3}sin x cos x
in two different ways. Give the answer with everything multiplied out (instead of factoring out common factors).