CAIE A-Level Mathematics 2.4 Differentiation Question Bank

Practise differentiating exponential, logarithmic and trigonometric functions explicitly, implicitly or parametrically and applying gradients to tangents and stationary points.

Exam points

apply product, quotient and chain rules to exponential, logarithmic and trig composites
differentiate both sides implicitly and collect every dy/dx term before solving
form dy/dx = (dy/dt)/(dx/dt) parametrically and evaluate gradients or stationary points

Covered syllabus points

2.4.1—Advanced differentiation 2.4.2—Product and quotient rules 2.4.3—Differentiation rules

[Maximum number: 3]

Figure for Question 2.4 Differentiation question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}$ . The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

Find the gradient of the curve at A.

Differentiate to obtain form $k_{1} \mathrm{e}^{-x}+k_{2} \mathrm{e}^{2 x}$

M1

Where $k_{1} k_{2} \neq 0, k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $-8 \mathrm{e}^{-x}-2 \mathrm{e}^{2 x}$

A1

Substitute x=0 to obtain -10

A1

[Maximum number: 3]

Figure for Question 2.4 Differentiation question 2 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\frac{2 \ln x}{3 x+1}$ . The curve crosses the x-axis at the point A and has a maximum point B. The shaded region is bounded by the curve and the lines x=3 and y=0.

Find the gradient of the curve at A.

Differentiate using quotient rule (or product rule)

M1

Obtain $\frac{(3 x+1) \frac{2}{x}-6 \ln x}{(3 x+1)^{2}}$

A1

OE

Substitute x=1 to obtain $\frac{1}{2}$

A1

OE

[Maximum number: 3]

Figure for Question 2.4 Differentiation question 3 — 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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Covered syllabus points

2.4 Differentiation question 1

2.4 Differentiation question 2

2.4 Differentiation question 3