At a Glance - Derivative of a Product of Functions
There's a convenient rule we can use to find the derivative of expressions like (3x)(sin x) or x^{2}e^{x}.
For the first expression, we know how to find the derivatives of 3x and sin x, but we don't yet have a way to find the derivative of their product. That's where the product rule comes in.
We use the product rule whenever there's 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 '.
Unfortunately, the derivative of a product is not the product of the derivatives. It's not like the derivative of the sum of two functions.
Product Rule
The derivative of f(x) · g(x) is f ' (x) · g(x) + f ' (x) · g(x).
In words, if we have a product of two functions, then the derivative of the product is
"derivative of the first times the second plus the first times the derivative of the second."
The examples will give you plenty of practice actually using the product rule.
Example 1
Find the derivative of h(x) = x^{2}e^{x}. |
Example 2
What is the derivative of f(x) = sin x cos x? |
Example 3
What's the derivative of g(x) = x^{2}sin x? |
Example 4
Find the derivative of f(x) = 4x^{2}. |
Example 5
What's 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 the following function.
- f(x) = x sin x
Exercise 7
What's the derivative of f(x)?
- f(x) = e^{x}cos x
Exercise 8
What's the derivative of the following function?
- f(x) = x ln x
Exercise 9
What's g ' (x) for the following function?
- g(x) = 5^{x}e^{x}
Exercise 10
What's the derivative of g(x)?
- g(x) = (log_{2 }x)(log_{3 }x)
Exercise 11
What's the derivative of g(x)?
- g(x) = 5e^{x}
Exercise 12
Find the derivative of the following function.
- h(x) = (x^{2} + 2x)ln x
Exercise 13
What is h ' (x) for the following function?
- h(x) = ln x cos x
Exercise 14
Find the derivative of h(x).
Exercise 15
Find the derivative of j(x).
- j(x) = ln x + cos x
Exercise 16
What is the derivative of f(x) = (x^{2} + 2)(x^{3} – 4).
- Use the product rule.
Exercise 17
What's the derivative of f(x) = (x^{2} + 2)(x^{3}– 4)?
- Rewrite f by multiplying the factors together, then take the derivative.
Exercise 18
- What's 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).