Differentiation Rules

The Quotient Rule

Find the derivative of a fraction of two functions using the quotient rule.

Example

Differentiate numerator and denominator, keeping the quotient-rule order.

highlighted = computed this step

Step 1 — Set up

Start with a quotient of two functions.

f(x)=sin⁡xx2+1f(x)= \frac{\sin x}{x^{2}+1}

Step 2 — Name numerator and denominator

Name the numerator u and denominator v.

u=sin⁡xv=x2+1u= \hlmath{\sin x} \quad v= \hlmath{x^{2}+1}

Step 3 — Differentiate parts

Differentiate u and v separately.

u′=cos⁡xv′=2xu'= \hlmath{\cos x} \quad v'= \hlmath{2x}

Step 4 — Quotient rule order

Use the quotient-rule order u prime v minus u v prime.

u′v−uv′v2\frac{u'v-uv'}{v^{ 2 }}

Step 5 — Assemble result

Assemble f prime g minus f g prime over the denominator squared.

f′(x)=cos⁡x(x2+1)−sin⁡x⋅2x(x2+1)2f'(x)= \hlmath{\frac{\cos x(x^{2}+1)-\sin x\cdot 2x}{(x^{2}+1)^{2}}}
quotient-rule The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. If f(x) = u(x) / v(x), then f'(x) = (u'v - uv') / v².