The idea

The quotient rule

Differentiating a quotient y = u/v as (u'v - uv') / v² — getting the order in the numerator and the squared denominator right, and when to prefer rewriting as a product instead.

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Pure · Differentiation

The quotient rule

Differentiating a quotient y = u/v as (u'v - uv') / v² — getting the order in the numerator and the squared denominator right, and when to prefer rewriting as a product instead.

Why it works

A product in disguise

Differentiate

y = \dfrac{2x + 1}{x^2 - 3}

. It isn't a sum, it isn't a plain power, and the top and bottom refuse to divide out — a genuine quotient, and the last differentiation shape the course needs. The good news: a quotient

y = \dfrac{u}{v}

is just a product in disguise,

y = u\,v^{-1}

, so the rules you already own build this one — no faith required.

With

y = u v^{-1}

, the product rule gives

\frac{dy}{dx} = u'\,v^{-1} + u\cdot(-1)v^{-2}v' = \frac{u'}{v} - \frac{uv'}{v^2}.

Put both terms over the common denominator

v^2

\frac{dy}{dx} = \frac{u'v - uv'}{v^2}.

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