AP® Calculus AB · Unit 2 · Topic 2.9
The Quotient Rule
The quotient rule differentiates u/v: take (derivative of top times bottom, minus top times derivative of bottom), all over bottom squared. Order matters because subtraction isn't symmetric.
The quotient rule says: if f(x) = u(x)/v(x), then f'(x) = (u'v - uv') / v2. Notice the minus sign. That's why the rule cares which function is on top. Swap u and v and you get a different, wrong answer, because u'v - uv' is not the same as v'u - vu' unless u'v happens to equal uv'. The product rule has no such worry, since addition doesn't care about order, but subtraction does.
You can actually derive the quotient rule from the product rule, so it isn't a separate fact to memorize blindly. Write u/v as u times v−1. Apply the product rule: derivative is u'v−1 + u(-1)v−2v'. Multiply through by v2/v2 to clear the negative exponents, and you land exactly on (u'v - uv')/v2. Seeing this connection means one less rule to memorize from scratch, since it's really the product rule plus the chain rule applied to v−1.
For (x2+1)/(3x-4): let u = x2+1, so u' = 2x; let v = 3x-4, so v' = 3. The derivative is [2x(3x-4) - (x2+1)(3)] / (3x-4)2, which expands to (6x2 - 8x - 3x2 - 3)/(3x-4)2 = (3x2 - 8x - 3)/(3x-4)2. For sin(x)/x2: u' = cos(x), v' = 2x, giving [x2cos(x) - 2xsin(x)] / x4, which simplifies to [xcos(x) - 2sin(x)] / x3 after canceling one factor of x. This rule sits right alongside the product rule as the two tools for breaking apart functions built from multiplication or division, before the chain rule handles composition.
Key terms
- Quotient rule
- The rule (u/v)' = (u'v - uv')/v2 for differentiating one function divided by another.
- Numerator function (u)
- The function on top of the fraction, whose derivative is u'.
- Denominator function (v)
- The function on the bottom of the fraction, whose derivative is v', squared in the final denominator.
- Negative power rewrite
- Writing u/v as u times v−1 so the product rule and chain rule can derive the quotient rule.
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End of topic.