Quotient Rule: Step-by-Step Guide
The quotient rule differentiates a ratio of two functions: if y = u(x)/v(x), then y' = (u'v - u v')/v^2. The order matters: low D-high minus high D-low, all over the square of low.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Identify numerator and denominator. Write the function as u(x)/v(x). The numerator is the 'high' and the denominator is the 'low'.\[y = \frac{u(x)}{v(x)}\]
- Differentiate top and bottom separately. Compute u'(x) and v'(x). Each may require its own rules (power, chain, etc.).\[u'(x), \;\; v'(x)\]
- Apply the quotient rule formula. Use y' = (u'v - u v')/v^2. Memory device: 'low D-high minus high D-low, square the bottom and away we go.'\[\frac{dy}{dx} = \frac{u'(x)\,v(x) - u(x)\,v'(x)}{[v(x)]^2}\]
- Simplify the result. Expand the numerator carefully (watching the minus sign), combine like terms, and factor where possible. Keep the v^2 in the denominator.\[\frac{dy}{dx} = \text{simplified}\]
Worked example
Find d/dx of y = (2x + 1)/(x^2 + 3).
Step 1: Identify numerator u = 2x + 1 and denominator v = x^2 + 3.
Step 2: Differentiate: u' = 2 and v' = 2x.
Step 3: Apply the quotient rule: y' = (u'v - uv')/v^2 = (2(x^2 + 3) - (2x + 1)(2x))/(x^2 + 3)^2.
Step 4: Expand the numerator: 2x^2 + 6 - (4x^2 + 2x) = 2x^2 + 6 - 4x^2 - 2x = -2x^2 - 2x + 6.
Step 5: Factor: -2(x^2 + x - 3). So y' = -2(x^2 + x - 3)/(x^2 + 3)^2.
Verification: At x = 0, y' = -2(0 + 0 - 3)/(9) = 6/9 = 2/3. Numerical: (y(0.001) - y(0))/0.001 approximately 0.6664, close to 2/3.
Tips and common mistakes
- Memorize 'low D-high minus high D-low, square the bottom and away we go' to keep the order straight.
- The minus sign matters — the order of u'v and u v' is NOT symmetric.
- Watch for sign errors when distributing the minus sign across u v'.
- If the numerator is a constant, the quotient rule simplifies to -c v'/v^2.
- Sometimes rewriting u/v as u * v^(-1) and using product + chain rules is cleaner.
- Always square the denominator — never forget the v^2 in the result.
- Combine quotient with chain rules when u or v is a composition.
- After simplifying, double-check by differentiating with an alternative method or numerically.