Product and Quotient Rules
Formulas for differentiating products and quotients of functions.
Includes worked examples for each rule.
The Product Rule
When you multiply two functions and need the derivative, use the product rule. A helpful mnemonic: "first times derivative of second, plus second times derivative of first."
The Quotient Rule
When you divide two functions and need the derivative, use the quotient rule. A helpful mnemonic: "low d-high minus high d-low, all over low squared."
Variables
| Symbol | Meaning |
|---|---|
| f(x) | The first function (numerator in quotient rule) |
| g(x) | The second function (denominator in quotient rule) |
| f'(x) | Derivative of f(x) |
| g'(x) | Derivative of g(x) |
Example 1 — Product Rule
Find d/dx [x³ · sin(x)]
f(x) = x³, g(x) = sin(x)
f'(x) = 3x², g'(x) = cos(x)
= 3x² · sin(x) + x³ · cos(x)
= 3x²·sin(x) + x³·cos(x)
Example 2 — Product Rule with Exponential
Find d/dx [x² · e^x]
f(x) = x², g(x) = e^x
f'(x) = 2x, g'(x) = e^x
= 2x · e^x + x² · e^x
= e^x(2x + x²) = x·e^x(2 + x)
Example 3 — Quotient Rule
Find d/dx [x² / (x + 1)]
f(x) = x², g(x) = x + 1
f'(x) = 2x, g'(x) = 1
= [2x(x + 1) − x² · 1] / (x + 1)²
= [2x² + 2x − x²] / (x + 1)²
= (x² + 2x) / (x + 1)²
Example 4 — Quotient Rule with Trig
Find d/dx [sin(x) / x]
f(x) = sin(x), g(x) = x
f'(x) = cos(x), g'(x) = 1
= [cos(x) · x − sin(x) · 1] / x²
= [x·cos(x) − sin(x)] / x²
When to Use Each
- Product rule: When two functions are multiplied: f(x) · g(x)
- Quotient rule: When one function is divided by another: f(x) / g(x)
- Tip: You can often avoid the quotient rule by rewriting f/g as f · g⁻¹ and using the product rule + chain rule instead