CAIE A-Level Mathematics 2.4.3 Implicit and Parametric Differentiation

Practise differentiating implicitly or parametrically, isolating dy/dx and using its value or sign to calculate gradients, tangents, normals and monotonic behaviour.

Exam points

differentiate every y-term implicitly with a dy/dx factor and collect those terms
calculate dx/dt and dy/dt separately before forming dy/dx = (dy/dt)/(dx/dt)
substitute the stated point or parameter and use the gradient for a tangent, normal or sign test

[Maximum number: 3]

Figure for Question 2.4.3—Differentiation rules question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with parametric equations

x=4 \mathrm{e}^{2 t}, \quad y=5 \mathrm{e}^{-t} \cos 2 t

for $-\frac{1}{4} \pi \leqslant t \leqslant \frac{1}{4} \pi$ . The curve has a maximum point M.

Find an expression for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of t.

Use product rule to find $\frac{\mathrm{d} y}{\mathrm{~d} t}$

M1

Obtain $\frac{\mathrm{d} y}{\mathrm{~d} t}=-5 \mathrm{e}^{-t} \cos 2 t-10 \mathrm{e}^{-t} \sin 2 t$

A1

or (unsimplified) equivalent (do not condone poor use of
brackets.

Obtain $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{-5 \mathrm{e}^{-t} \cos 2 t-10 \mathrm{e}^{-t} \sin 2 t}{8 \mathrm{e}^{2 t}}$

A1

OE following their expression for $\frac{\mathrm{d} y}{\mathrm{~d} t}$ .

3

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2.4.3—Differentiation rules question 1