Quotient Rule

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 $v(x)\ne0$,

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

Why the denominator is squared

Write $u/v=u\cdot v^{-1}$ and combine the product and chain rules:

This derivation explains both the subtraction and squared denominator.

Worked example

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

A reliable layout

Before substituting, write

Then place them into $u'v-uv'$ with full parentheses. This reduces sign and omission errors, especially when $u$ or $v'$ has multiple terms.

Simplify before differentiating

If a quotient can be rewritten as powers or simplified without changing the relevant domain, basic rules may be easier. For example,

Its derivative is $2x$ on the original domain $x\ne0$. The simplified formula must not erase the hole from the original function.

Quotient with chain rule

For

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

Factoring common powers can simplify the result.

Evaluating from a table

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

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

Motion and average quantities

Quotients arise in average cost, density, concentration, and efficiency. If total cost $C(q)$ is divided by quantity $q$, then average cost is $A(q)=C(q)/q$ and

The result compares marginal and average behaviour.

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.

Quick self-check

• Is the original expression truly a quotient that cannot be simplified more easily?
• Have $u,u',v,v'$ been identified?
• Is the numerator $u'v-uv'$ in that order?
• Is the complete denominator squared?
• Are inner chain rules and domain restrictions included?
• Can substitution or an alternative form verify the result?

Related topics

• Product Rule
• Chain Rule
• Rational Functions
• Basic Differentiation Rules