Differentiation summary

Key learning points

• $\frac{d}{dx}(ax^n)=anx^{n-1}$; constants differentiate to 0.
• Stationary points where $\frac{dy}{dx}=0$; classify with the second derivative when useful.
• Tangent gradient equals $\frac{dy}{dx}$ at the point; the normal is perpendicular.
• Increasing/decreasing follows the sign of $\frac{dy}{dx}$.

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.

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.