Product Rule
Derivative of a product of two functions.
Formula
Variables
Example
d/dx (x²·sin x) = 2x·sin x + x²·cos x.
How it works
When two functions are multiplied, the derivative is not just "differentiate the first and keep the second" or vice versa; both parts change at the same time, so both contributions have to be included. The product rule comes from the limit definition of the derivative: if you compare the change in f(x)g(x) to the original product, the variation splits into one part from the change in f and one part from the change in g. That is why the result is f'g + fg'.
When to use it
Use this any time you need the slope or rate of change of a product of two x-dependent quantities, such as area formulas, physics expressions, or functions like x^n times trig/exponential terms. It is especially useful in calculus problems where one factor is easy to differentiate but the other is also changing with x.
Worked example, step by step
Differentiate h(x) = (3x^2 - 1)e^x Let f(x) = 3x^2 - 1 and g(x) = e^x Then f'(x) = 6x and g'(x) = e^x Apply the product rule: h'(x) = f'g + fg' So h'(x) = (6x)(e^x) + (3x^2 - 1)(e^x) Combine like terms: h'(x) = e^x(6x + 3x^2 - 1) Final answer: h'(x) = e^x(3x^2 + 6x - 1)
Common mistakes to avoid
- only differentiating the first factor and forgetting the second term
- treating (fg)' as f'g' instead of f'g + fg'
- not applying the chain rule too when one factor is itself a composite function
Did You Know?
The product rule is NOT just the product of the derivatives — a mistake Leibniz himself first made.
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