Differentiation 2 summary

Key learning points

• Product $(uv)'=u'v+uv'$; quotient $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$.
• Chain rule: differentiate outside, multiply by inside derivative.
• Standard derivatives: $\sin$, $\cos$, $e^{kx}$, $\ln x$.
• Implicit differentiation when $y$ is not isolated.

Worked example 1 — Differentiate

Differentiate $y=3x^4-2x$.

Working

Power rule.

Worked example 2 — Gradient at a point

$y=x^2$ at $x=3$. Gradient?

Working

Differentiate, then substitute.

Worked example 3 — Stationary point

$y=x^2-4x+1$. Stationary $x$?

Working

Set derivative to zero.

Worked example 4 — Product rule

Differentiate $y=x^2\sin x$.

Working

Keep the order $u'v+uv'$.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.