CAIE A-Level Mathematics A2 3.4 Differentiation Questions

Practise differentiating exponential, logarithmic, trigonometric, inverse-tangent, implicit and parametric curves and applying derivatives to exact gradients and stationary points.

Exam points

combine chain, product and quotient rules for exponential, logarithmic and trig forms
differentiate implicit relations and collect dy/dx terms before substituting a point
form parametric dy/dx from dy/dt and dx/dt and solve gradient or normal conditions

Covered syllabus points

3.4.1—Advanced differentiation 3.4.2—Product and quotient rules 3.4.3—Differentiation rules

[Maximum number: 1]

The variables x and $\theta$ satisfy the differential equation

x \sin ^{2} \theta \frac{\mathrm{~d} x}{\mathrm{~d} \theta}=\tan ^{2} \theta-2 \cot \theta

for $0<\theta<\frac{1}{2} \pi$ and x>0. It is given that x=2 when $\theta=\frac{1}{4} \pi$ .

Show that $\frac{\mathrm{d}}{\mathrm{d} \theta}\left(\cot ^{2} \theta\right)=-\frac{2 \cot \theta}{\sin ^{2} \theta}$ .
(You may assume without proof that the derivative of $\cot \theta$ with respect to $\theta$ is $-\operatorname{cosec}^{2} \theta$ .)

Show sufficient working to justify the given statement
e.g. see $2 \cot \theta \times-\operatorname{cosec}^{2} \theta$ in the working
or express in terms of $\sin \theta$ and $\cos \theta$ and use quotient
rule to obtain the given result. Solution must have $\theta$
present throughout and must reach the given answer.

[Maximum number: 4]

Figure for Question 2 — CAIE A-Level Mathematics A2

The diagram shows the curve $y=x \mathrm{e}^{-\frac{1}{4} x^{2}}$ , for $x \geqslant 0$ , and its maximum point M.

Find the exact coordinates of M.

Use the correct product rule

*M1

Condone error in chain rule.

Obtain correct derivative in any form
e.g. $\frac{d y}{d x}=-\frac{x^{2}}{2} e^{-\frac{x^{2}}{4}}+e^{-\frac{x^{2}}{4}}$ .

Equate derivative to zero and solve for x
Obtain answer $\left(\sqrt{2}, \sqrt{2} \mathrm{e}^{-\frac{1}{2}}\right)$
Or exact equivalent.
Can state the components separately.

[Maximum number: 8]

The equation of a curve is $2 y^{3}-3 x^{2} y-x^{3}=16$ ．

(a)

Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x^{2}+2 x y}{2 y^{2}-x^{2}}$ ．

[ 4 ]

(b)

Hence find the coordinates of the points on the curve at which the normal is parallel to the y-axis. [4]

[ 4 ]

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Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)