CAIE A-Level Mathematics 1.7 Differentiation Question Bank

CAIE A-Level Mathematics 1.7 Differentiation Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise differentiating powers and composite expressions and applying derivatives to gradients, tangents, normals, stationary points, monotonicity and connected rates.

Exam points

  • apply power and chain rules and simplify first or second derivatives into usable form
  • set dy/dx to a gradient or zero to find tangents, normals and stationary points
  • connect rates with dy/dt = (dy/dx)(dx/dt) and substitute values with consistent units

Covered syllabus points

1.7.1—Gradient as limit1.7.2—Differentiation rules1.7.3—Differentiation applications1.7.4—Stationary points

Question 3

[Maximum number: 2]

The equation of a curve is y=f(x), where f(x)=12x23(x−2)2\mathrm{f}(x)=\frac{1}{2} x^{\frac{2}{3}}(x-2)^{2}. The following points lie on the curve. Non-exact values of the y-coordinates are given correct to 6 decimal places.

A(8,72), B(8.001, k), C(8.01,72.300388), D(8.1,75.038882)

Question 3(b)

(a)

Find the gradient of the chord AD. Give your answer correct to 4 decimal places.

[ 1 ]

Question 3(c)

(b)

State what the values in the table suggest about the value of f'(8).

[ 1 ]

Question 11

[Maximum number: 6]

A curve passes through the point P(4,3) and is such that

dy dx=8x2−10(2x−3)2.\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{8}{x^{2}}-\frac{10}{(2 x-3)^{2}} .

Question 11(a)

(a)

Find the equation of the normal to the curve at P. Give your answer in the form y=m x+c.

[ 3 ]

Question 11(b)

(b)

Find the rate of change of the gradient of the curve when x=4.

[ 3 ]

Question 11

[Maximum number: 13]

The equation of a curve is y=83x−8−6x−1y=\frac{8}{3 x-8}-\frac{6}{x-1}.

Question 11(a)

(a)

Find the coordinates of the point at which the tangent to the curve at the point (3,5) intersects the line y=-8 x.

[ 6 ]

Question 11(b)(i)

(b)

（b）（i）Find the x－coordinates of each of the stationary points of the curve．

[ 3 ]

Question 11(b)(ii)

(c)

（ii）Find d2y dx2\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}} and hence determine the nature of each of the stationary points．

[ 4 ]

Question 11(a)

[Maximum number: 5]
Figure for Question 11(a) — CAIE A-Level Mathematics

The diagram shows the curve with equation y=4x2−x3y=4 x^{2}-x^{3} and the tangent to the curve at the point P. The point P has x-coordinate 3.

Find the equation of the tangent to the curve at the point P. Give your answer in the form y=m x+c.