Product Rule — Mathematics Knowledge System

Overview

The product rule gives the derivative of a product of two functions. It is not simply the product of the derivatives — a common error — but rather a sum of two terms, each holding one factor fixed while differentiating the other.

Interactive Graph

Formal Definition

Definition

For differentiable functions f and g:

\frac{d}{dx}\big[f(x)g(x)\big] = f'(x)g(x) + f(x)g'(x)

Product Rule

Worked Examples

Let f(x)=x², g(x)=sin(x); f'(x)=2x, g'(x)=cos(x).
$f'=2x,\ g'=\cos x$
Apply the product rule: h' = f'g + fg'.
$h'(x) = 2x\sin(x) + x^2\cos(x)$

Answer: h'(x) = 2x sin(x) + x² cos(x)

Practice Problems

Difficulty 3/10

Differentiate f(x) = x³ ln(x).

Difficulty 4/10

Differentiate f(x) = (x² + 1)(x³ - 2x).

Common Mistakes

Common Mistake

Differentiating a product by simply multiplying the two derivatives: (fg)' = f'g'.

This is false in general — the correct rule is (fg)' = f'g + fg'. Counterexample: f=g=x gives (x²)'=2x, but f'g'=1·1=1≠2x.

Summary

Product rule: (fg)' = f'g + fg' — differentiate one factor at a time, keeping the other fixed.
It generalizes to three or more factors by applying it repeatedly.
A frequent error is multiplying the derivatives together; always check with a simple example like f=g=x.

References

WebsiteWikipedia — Product rule