Product and Quotient Rules
Reference for the product rule (fg)' = fg'+gf' and quotient rule (f/g)' = (gf'-fg')/g^2.
Side-by-side worked examples and tips for choosing the right 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