The Product Rule: How to Differentiate a Product of Functions
Learn the product rule in two steps with two worked examples and the most common mistake.
The rule
- d/dx [f(x) · g(x)] = f′(x) · g(x) + f(x) · g′(x)
Derivative of the first times the second, plus the first times the derivative of the second.
Example 1
Differentiate y = x² · sin x
- f = x², f′ = 2x
- g = sin x, g′ = cos x
- 2x · sin x + x² · cos x
2x sin x + x² cos x
Example 2
Differentiate y = (3x + 1)(x − 4)
- f = 3x + 1, f′ = 3
- g = x − 4, g′ = 1
- 3(x − 4) + (3x + 1)(1)
6x − 11
Common mistake
d/dx [x² · sin x] = 2x · cos x
The derivative of a product is not the product of the derivatives.
You need both terms: f′g and fg′.
Want more Calculus practice?
The free Calculus study guide has quizzes, flashcards and practice exams. 100% free, no sign-up needed.
Open the Calculus guide