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: 3]

The equation of a curve is $y=\tan ^{-1}(4 x)$ .

Find the exact values of x when the gradient of the curve is $\frac{1}{4}$ .

Differentiate to obtain $\frac{k_1}{1+Bx^2}$ or $\frac{k_2}{x^2+\frac1B}$ . M1

Obtain
$\frac{dy}{dx}=\frac4{1+16x^2}.$
Set $\frac{dy}{dx}=\frac14$ and obtain
$x=\pm\frac{\sqrt{15}}4.$
Accept exact equivalents, including $\pm\frac{\sqrt{15}}{\sqrt{16}}$ .

Alternative method: rewrite as $\tan y=4x$ , use $\sec^2y=k\frac{dx}{dy}$ , and obtain the same exact values.

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