Parametric Functions Topic Review summary

Key learning points

• $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ when $\frac{dx}{dt}\neq0$.
• Second derivative: differentiate $\frac{dy}{dx}$ w.r.t. $t$, then divide by $\frac{dx}{dt}$.
• Eliminate $t$ for a Cartesian equation when asked.
• Tangents/normals at a given parameter use the parametric gradient.
• Domain of $t$ matters for sketches and intersections.

Worked example 1 — Gradient

$x=t^2$, $y=t^3$ at $t=2$.

Working

$\frac{3t}{2}$.

Worked example 2 — Eliminate

$x=2t$, $y=t^2$.

Working

$t=x/2$.

Worked example 3 — \dy{dx}=\dy/dt{dx/dt} when \dx{dt}\\neq0

$\\frac{dy}{dx}=\\frac{dy/dt}{dx/dt}$ when $\\frac{dx}{dt}\\neq0$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Second derivative: differentiate \dy{dx} w

Second derivative: differentiate $\\frac{dy}{dx}$ w.r.t. $t$, then divide by $\\frac{dx}{dt}$.

Working

Attempt without notes, then check the answers PDF.

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.