Quotient Rule

The quotient rule differentiates one changing function divided by another while preserving the denominator's domain restriction.

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Identify structure → choose rule → verify. A formula or convergence question is inspected for structure and conditions, routed to an appropriate calculus rule or test, and checked against units, endpoints, differentiation, or the stated hypotheses.

A quotient's derivative balances numerator change against denominator change over the square of the denominator.

The rule

For differentiable $u(x)$ and $v(x)$ with $v(x)\ne0$,

\frac{d}{dx}\left[\frac{u}{v}\right] =\frac{u'v-uv'}{v^2}.

Order matters in the numerator: derivative of the top times the bottom minus the top times derivative of the bottom.

Worked example

The original domain restriction remains part of the derivative's context.

Quotient with chain rule

For

y=\frac{\sin x}{(x^2+1)^3},

the quotient rule handles the division, while the chain rule differentiates the denominator:

y'=\frac{\cos x(x^2+1)^3-\sin x\cdot6x(x^2+1)^2}{(x^2+1)^6}.

Factoring common powers can simplify the result.

Evaluating from a table

If $f=u/v$, then at $x=a$:

f'(a)=\frac{u'(a)v(a)-u(a)v'(a)}{[v(a)]^2}.

All four function/derivative values are needed, and $v(a)$ must be nonzero.

Finding critical numbers

For a rational derivative, candidates for horizontal tangents come from zeros of the derivative numerator, while values making the original function undefined are not critical numbers in its domain.

Analyze the factored derivative and keep domain exclusions separate.

Common mistakes

Using $u'v+uv'$. That is the product rule; quotient uses subtraction and $v^2$.

Reversing the numerator order. Keep $u'v-uv'$.

Squaring only part of the denominator. Square the complete $v(x)$.

Forgetting a chain rule inside $u'$ or $v'$. Differentiate each function fully.

Restoring a cancelled excluded value. Preserve the original domain.