CAIE A-Level Mathematics AS 2.4 Differentiation Questions

Practise differentiating exponential, logarithmic and trigonometric functions explicitly, implicitly or parametrically and applying gradients to tangents and stationary points.

Exam points

apply product, quotient and chain rules to exponential, logarithmic and trig composites
differentiate both sides implicitly and collect every dy/dx term before solving
form dy/dx = (dy/dt)/(dx/dt) parametrically and evaluate gradients or stationary points

Covered syllabus points

2.4.1—Advanced differentiation 2.4.2—Product and quotient rules 2.4.3—Differentiation rules

[Maximum number: 3]

Figure for Question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}$ . The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

Find the gradient of the curve at A.

Differentiate to obtain form $k_{1} \mathrm{e}^{-x}+k_{2} \mathrm{e}^{2 x}$
Where $k_{1} k_{2} \neq 0, k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $-8 \mathrm{e}^{-x}-2 \mathrm{e}^{2 x}$
Substitute x=0 to obtain -10

[Maximum number: 4]

Figure for Question 2 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\frac{\ln (2 x+1)}{x+3}$ . The curve has a maximum point M.

(a)

Find an expression for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

[ 2 ]

Attempt use of quotient rule
Or equivalent method.

Obtain $\frac{(x+3) \frac{2}{2 x+1}-\ln (2 x+1)}{(x+3)^{2}}$
OE

(b)

Show that the x-coordinate of M satisfies the equation $x=\frac{x+3}{\ln (2 x+1)}-0.5$ .

[ 2 ]

Equate first derivative to zero and arrange as far as $2 x+1=\ldots$
Confirm $x=\frac{x+3}{\ln (2 x+1)}-0.5$
Answer given - necessary detail needed.

[Maximum number: 7]

A curve is defined by the parametric equations

x=4 \cos ^{2} t, \quad y=\sqrt{3} \sin 2 t,

for values of t such that $0<t<\frac{1}{2} \pi$ .
Find the equation of the normal to the curve at the point for which $t=\frac{1}{6} \pi$ . Give your answer in the form a x+b y+c=0 where a, b and c are integers.

Obtain forms $\frac{\mathrm{d} x}{\mathrm{~d} t}=k_{1} \cos t \sin t$ or $\frac{\mathrm{d} y}{\mathrm{~d} t}=k_{2} \cos 2 t$

*M1

Obtain correct $-8 \cos t \sin t$ or $-4 \sin 2 t$ and $2 \sqrt{3} \cos 2 t$
Attempt value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when $t=\frac{1}{6} \pi$

*DM1

Need to see attempt at substitution.

Obtain $\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{1}{2}$
State or imply gradient of normal is 2

**M1 FT

Following their value of the first derivative.

Attempt equation of normal

**DM1

Not tangent and with attempt to find coordinates $\left(3, \frac{3}{2}\right)$ .

Obtain 4 x-2 y-9=0
Or equivalent of requested form.

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3