CAIE A-Level Mathematics A2 3.4.1 Advanced Differentiation Questions

Practise advanced differentiation techniques for composite, exponential and trigonometric functions.

Exam points

differentiate composite, exponential and trigonometric functions
apply product and quotient rules in mixed expressions
use derivatives to analyse gradients and rates of change

[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.

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CAIE A-Level Mathematics A2 3.4.1 Advanced Differentiation Questions question 1