A Level · Mathematics 9709 · Pure Mathematics 3

Differentiation: A Level Mathematics questions with answers

Free Differentiation questions for A Level Mathematics (9709), part of Pure Mathematics 3. Each question has the answer and the marking points an examiner would look for.

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Differentiation practice questions

Question 1Structured · 9 marksA* challenge

A curve has equation x2 + xy + y2 = 12.

(a)

Show that dydx = −2x + yx + 2y.

[3]
(b)

Find the coordinates of the points on the curve where the tangent is parallel to the x-axis.

[4]
(c)

Find the coordinates of the points where the tangent is parallel to the y-axis.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2x + y + xdydx + 2ydydx = 0
implicit differentiation of y2 B1
product rule on xy M1
rearranged correctly A1
3
(b)(2, −4) and (−2, 4)
2x + y = 0 M1
substitute y = −2x into the curve M1
3x2 = 12 A1
(2, −4) and (−2, 4) A1
4
(c)(−4, 2) and (4, −2)
x + 2y = 0 used M1
both points A1
2
Question 2Structured · 13 marksA* challenge

A curve is defined by the parametric equations x = 3t2, y = 2t3. The point P on the curve has parameter p, where p ≠ 0.

(a)

Show that the equation of the tangent at P is y = px − p3.

[2]
(b)

The tangent at P meets the curve again at Q. Find the parameter of Q in terms of p.

[3]
(c)

Find the values of p for which the tangent at P is also the normal to the curve at Q.

[2]
(d)

For the positive value of p, find the exact length of PQ.

[2]
(e)

For the positive value of p, find the exact area of the region enclosed by the line PQ and the curve.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)dydx = 6t26t = t
dydx = 6t26t = t B1
y − 2p3 = p(x − 3p2) simplified A1
2
(b)t = −p2
substitute x = 3t2, y = 2t3 into the tangent: 2t3 − 3pt2 + p3 = 0 M1
t = p is a repeated root: (t − p)2(2t + p) = 0 M1
t = −p2 A1
3
(c)p = ±√2
gradient at Q is −p2, so p × (−p2) = −1 M1
p = ±√2 A1
2
(d)9√32
P(6, 4√2), Q(32, −√22) M1
PQ = 9√32 A1
2
(e)81√240
area under curve = ∫ y dxdt dt = ∫ 12t4 dt with limits −√22 and √2 (signed) M1
125t5 evaluated: 125(4√2 + √28) = 99√210 A1
area under the line PQ between x = 32 and x = 6 (trapezium, signed) = 63√28 M1
99√210 − 63√28 = 81√240 A1
4
Question 3Structured · 4 marks

A curve has equation x2 + 3xy + y2 = 5. Find the gradient of the curve at the point (1, 1).

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
−1
implicit differentiation M1
3y + 3xdydx product rule A1
2x + 3y + (3x + 2y)dydx = 0 A1
−1 A1
4
Question 4Structured · 4 marks

Find the gradient of the curve y = x2e3x at the point where x = 1. Give your answer correct to 3 significant figures.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
100
Product rule attempted M1
2xe3x + 3x2e3x A1
Substitute x = 1 M1
100 A1
4
Question 5Structured · 6 marksA* challenge

A curve has equation y = ln xx3 for x > 0.

(a)

Find the exact coordinates of the stationary point.

[4]
(b)

Determine the nature of the stationary point.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(e1/3, 13e)
quotient rule M1
1 − 3ln xx4 A1
ln x = 13 M1
(e1/3, 13e) A1
4
(b)maximum: gradient changes from positive to negative
valid method (sign of derivative either side or second derivative) M1
maximum A1
2
Question 6Structured · 9 marksA* challenge

A curve has equation x2 + xy + y2 = 3.

(a)

Show that dydx = −2x + yx + 2y.

[3]
(b)

Find the coordinates of the points on the curve where the tangent is parallel to the x-axis.

[4]
(c)

Find the coordinates of the points where the tangent is parallel to the y-axis.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2x + y + xdydx + 2ydydx = 0
implicit differentiation of y2 B1
product rule on xy M1
rearranged correctly A1
3
(b)(1, −2) and (−1, 2)
2x + y = 0 M1
substitute y = −2x into the curve M1
3x2 = 3 A1
(1, −2) and (−1, 2) A1
4
(c)(−2, 1) and (2, −1)
x + 2y = 0 used M1
both points A1
2

More Pure Mathematics 3 questions

AlgebraLogarithmic & exponential functionsTrigonometryIntegrationNumerical solution of equationsVectorsDifferential equationsComplex numbers

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in