Product, Quotient and Chain Rules

≈ 25 min

Three rules let you differentiate combinations of functions.

Product rule: $\dfrac{d}{dx}[u\,v] = u'v + uv'$ .

Quotient rule: $\dfrac{d}{dx}\!\left[\dfrac{u}{v}\right] = \dfrac{u'v - uv'}{v^2}$ .

Chain rule (a function inside a function): $\dfrac{d}{dx}f\big(g(x)\big) = f'\big(g(x)\big)\cdot g'(x)$ .

Worked example (chain rule). Differentiate $y=(2x+1)^3$ .

The outer function is 'cube it'; the inner function is $2x+1$ . Differentiate the outer, keep the inner, then multiply by the inner's derivative:

\frac{dy}{dx} = 3(2x+1)^2 \cdot 2 = 6(2x+1)^2

Worked example (product rule). Differentiate $y=(x^2+1)(x-3)$ with $u=x^2+1$ , $v=x-3$ :

y' = (2x)(x-3) + (x^2+1)(1) = 2x^2-6x+x^2+1 = 3x^2-6x+1

Which rule is the efficient tool to differentiate $y=(3x+2)^5$ ?

Product rule

Quotient rule

Chain rule

No rule needed — it is already a simple power

Differentiate $y=(2x+1)^3$ . Give $\dfrac{dy}{dx}$ in factored form.

Differentiate $y=(x^2+1)(x-3)$ . Give $\dfrac{dy}{dx}$ as an expanded polynomial.

For $y=(x^2+1)^4$ , find $\dfrac{dy}{dx}$ at $x=1$ .