CAIE A-Level Mathematics 3.4.1 Advanced Differentiation

Practise differentiating eˣ, ln x, trigonometric and inverse-tangent composites and applying exact derivative equations to find gradients, coordinates and stationary points.

Exam points

differentiate eᵘ, ln u, tan u or tan⁻¹u and retain the derivative of the inner function
combine advanced derivatives with product or quotient rules before simplifying
set dy/dx to zero or a stated gradient and solve for exact coordinates or parameters

[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
$\frac{dy}{dx}=\frac4{1+16x^2}.$
A1

Set $\frac{dy}{dx}=\frac14$ and obtain
$x=\pm\frac{\sqrt{15}}4.$
A1
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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3.4.1—Advanced differentiation question 1