The Product Rule
Simple Explanation
To differentiate a PRODUCT of two functions, you cannot just multiply their derivatives. Instead: (f·g)' = f'·g + f·g' — the derivative of the first times the second, plus the first times the derivative of the second.
Why Do We Need It?
Many real expressions are naturally products of two changing quantities (like price × quantity), and this rule is the only correct way to differentiate them directly.
Formula
The Product Rule
(f·g)' = f'·g + f·g'
The derivative of a product of two functions is the derivative of the first times the second, plus the first times the derivative of the second.
- f, g
- — two differentiable functions of x
When to use it: Whenever two functions of x are multiplied together — never just multiply their individual derivatives.
Worked Example
Apply the Product Rule
Differentiate f(x) = x²·(x+3), using the Product Rule.
Why Does This Work?
Expanding f(x)=x²(x+3)=x³+3x² directly and differentiating with the power rule gives f'(x)=3x²+6x — exactly matching the Product Rule's result, confirming the rule is consistent with what you already know. The rule itself comes from carefully expanding the limit definition for a product, adding and subtracting a helpful term (u(x)v(x+h) − u(x)v(x+h)) to split it into two separate limits that become u'v and uv'.
Real-Life Example
How total revenue changes when both price and quantity change
A company's total revenue is price × quantity sold, and BOTH price and quantity are changing over time (e.g. as a promotion runs).
The Product Rule is needed to correctly find how revenue changes overall, since simply multiplying the two individual rates of change would be wrong.
Practice
Differentiate f(x) = x·(x+2) using the Product Rule, then evaluate f'(3).
HardCommon mistake
Assuming (f·g)' = f'·g' (just multiplying the two derivatives together) — this is NOT the Product Rule and generally gives the wrong answer.
Quick Review
- (f·g)' = f'·g + f·g'.
- Never just multiply the two derivatives together.
- Useful whenever two functions of x are multiplied.