Calculus — Derivatives⏱ 5 min read

Product Rule

d/dx(fg) = fg' + gf'

What is the Product Rule?

The product rule differentiates a product of two functions. In words: "first times the derivative of the second, plus the second times the derivative of the first." It's easy to reach for the shortcut of just multiplying f' and g' together — that shortcut is wrong, which is why this rule exists.

For three functions multiplied together, the rule extends naturally: (fgh)' = f'gh + fg'h + fgh' — each term differentiates exactly one of the three functions and leaves the other two unchanged.

What Each Variable Means

First functionAny differentiable function of x.

Second functionAny differentiable function of x.

Derivative of fdf/dx — how f changes with x.

Derivative of gdg/dx — how g changes with x.

When to Use It

Differentiating any expression that's a product of two (or more) functions, like x²·sin x
As the basis for deriving the quotient rule (writing f/g as f·g⁻¹)
Anywhere two changing quantities are multiplied together, such as in related-rates problems

Step-by-Step Examples

Example 1: Product of a polynomial and a trig function

Problem: Differentiate y = x² · sin x

Identify f and g

Split the product into its two factors.

f = x², g = sin x

Find f' and g'

Differentiate each factor separately.

f' = 2x, g' = cos x

Apply the product rule: fg' + gf'

Combine according to the rule.

dy/dx = x²·cos x + sin x·2x

✓

Answer: dy/dx = x²cos x + 2x sin x

Example 2: Product of two exponential-type functions

Problem: Differentiate y = eˣ · ln x

Identify f and g

Split the product into its two factors.

f = eˣ, g = ln x

Find f' and g'

Differentiate each factor separately.

f' = eˣ, g' = 1/x

Apply the product rule: fg' + gf'

Combine according to the rule.

dy/dx = eˣ·(1/x) + ln x·eˣ

✓

Answer: dy/dx = eˣ(1/x + ln x)

Common Mistakes

Mistake: Multiplying the two derivatives together: d/dx(fg) = f'g'.
Fix: That shortcut is incorrect. The correct rule is d/dx(fg) = f·g' + g·f' — each factor is paired with the other's derivative, then the two products are added.