The Quotient Rule: How to Differentiate a Fraction of Functions
Learn the quotient rule with two worked examples and the most common mistakes.
The rule
- d/dx [f/g] = (f′g − f g′) / g²
Low d-high minus high d-low, over the low squared. Order matters in the top.
Example 1
Differentiate y = x² / (x + 1)
- f = x², f′ = 2x; g = x + 1, g′ = 1
- (2x(x + 1) − x²(1)) / (x + 1)²
- Simplify the top: 2x² + 2x − x² = x² + 2x
(x² + 2x) / (x + 1)²
Example 2
Differentiate y = sin x / x
- f = sin x, f′ = cos x; g = x, g′ = 1
- (cos x · x − sin x · 1) / x²
(x cos x − sin x) / x²
Common mistake
d/dx [f/g] = f′ / g′
The derivative of a quotient is not the quotient of the derivatives.
Keep f′g first, then subtract fg′. Swapping them flips the sign.
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