CAIE A-Level Mathematics 2.4 Differentiation Question Bank

CAIE A-Level Mathematics 2.4 Differentiation Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

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 differentiation2.4.2—Product and quotient rules2.4.3—Differentiation rules

Question 6

[Maximum number: 4]
Figure for Question 6 — CAIE A-Level Mathematics

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

Question 6(a)

(a)

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

[ 2 ]

Question 6(b)

(b)

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

[ 2 ]

Question 6(b)

[Maximum number: 6]

A curve has equation (x2−3)ln⁡y+6x=14\left(x^{2}-3\right) \ln y+6 x=14.

Find the equation of the tangent to the curve at the point (2,e2)\left(2, \mathrm{e}^{2}\right). Give your answer in the form y=m x+c, where m and c are exact constants.

Question 8

[Maximum number: 6]

The equation of a curve is y=4e1−2x3x−1y=4 \mathrm{e}^{1-2 x} \sqrt{3 x-1}.
Find the exact coordinates of the stationary point of the curve.