Quotient rule

Objective. Differentiate quotients while preserving structure.

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Learn: the key idea

The quotient rule differentiates one function divided by another. If y = u/v, then dy/dx = (v(du/dx) − u(dv/dx))/v². The order in the numerator matters: the denominator's derivative term is subtracted, not added, and the whole expression is divided by v squared. For example, if y = x²/(x+1), with u = x² and v = x+1, dy/dx = ((x+1)(2x) − x²(1))/(x+1)². Always keep u and v in a fixed order and subtract carefully.

Worked example

Differentiate y = (2x)/(x² + 1) using the quotient rule.

1 $Let u = 2x and v = x² + 1, so du/dx = 2 and dv/dx = 2x$
2 $Apply dy/dx = (v(du/dx) - u(dv/dx))/v²$
3 $dy/dx = ((x²+1)(2) - 2x(2x))/(x²+1)²$
4 $Expand and simplify to dy/dx = (2 - 2x²)/(x²+1)²$

Answer: dy/dx = (2 − 2x²)/(x²+1)²

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Common mistake. Students often reverse the subtraction order, writing u(dv/dx) − v(du/dx) instead of v(du/dx) − u(dv/dx), which flips the sign. Fix this by always writing the denominator's derivative term first.

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