CAIE A-Level Mathematics 3.4 Differentiation Question Bank

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.

Use the correct product rule

*M1

Condone error in chain rule.

Obtain correct derivative in any form

A1

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

DM1

Obtain answer $\left(\sqrt{2}, \sqrt{2} \mathrm{e}^{-\frac{1}{2}}\right)$

A1

Or exact equivalent.
Can state the components separately.

State d x/ d t=2 t-2/2 t+1

B1

Use correct quotient rule or correct product rule

M1

Might split first as 1/2-1/4 t+2.

Obtain d y/ d t=(2 t+1). 1-t(2)(2 t+1)^2 oe

A1

Simplifies to 1(2 t+1)^2

Use d y/ d x=d y/ d t / d x/ d t

M1

Marking guidance:

Allow for d x/ d y.

Obtain answer 1/2(2 t+1)(2 t-1)(t+1)

A1

OE. Must have 1 in numerator.

May have, e.g., 1(2 t+1)(4 t^2+2 t-2). ISW.

Obtain x=(y/1-2 y)^2- (2 y/1-2 y+1)

B1

OE, e.g. x=(y/1-2 y)^2- (1/1-2 y).

Use correct quotient rule to differentiate

M1

Obtain d x/ d y=(2 y/1-2 y)((1-2 y)+2 y(1-2 y)^2)-2/1-2 y

A1

OE, d y/ d x=(1-2 y)^32 y-2(1-2 y)^2=(1-2 y)^3-2(1-4 y)(1-y)d y/ d x=(1-2 t/2 t+1)^3-2(1-4 t/2 t+1)(1-t/2 t+1)=1/-2(2 t+1)(1-2 t)(t+1)

M1

Express in terms of t.

Obtain answer 1/2(2 t+1)(2 t-1)(t+1)

A1

OE