CAIE A-Level Mathematics 3.4 Differentiation Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise differentiating exponential, logarithmic, trigonometric, inverse-tangent, implicit and parametric curves and applying derivatives to exact gradients and stationary points.

Exam points

combine chain, product and quotient rules for exponential, logarithmic and trig forms
differentiate implicit relations and collect dy/dx terms before substituting a point
form parametric dy/dx from dy/dt and dx/dt and solve gradient or normal conditions

Covered syllabus points

3.4.1—Advanced differentiation 3.4.2—Product and quotient rules 3.4.3—Differentiation rules

[Maximum number: 4]

A curve has equation $y=\frac{1+\mathrm{e}^{2 x}}{1+3 x}$ . The curve has exactly one stationary point P.

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ and hence show that the x-coordinate of P satisfies the equation $x=\frac{1}{6}+\frac{1}{2} \mathrm{e}^{-2 x}$ .

[Maximum number: 9]

A curve has parametric equations

x=\tan \theta, \quad y=\sin \theta-2 \sin ^{3} \theta,

for $0<\theta<\frac{1}{2} \pi$ .

(a)

Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=6 \cos ^{5} \theta-5 \cos ^{3} \theta$ .

[ 4 ]

(b)

（b）Find the equation of the normal to the curve at the point where it crosses the x－axis．Give your answer in the form y=m x+c ，where m and c are exact constants．

[ 5 ]

[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 $\frac{1}{4}$ .

Exam points

Covered syllabus points

Question 5(a)

Question 6

Question 6(a)

Question 6(b)

Question 7(a)