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AS A-level Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Differentiation

Chain, product and quotient rules, differentiating trig, exponential and log functions, implicit differentiation, rates of change, concave and convex curves, points of inflection.

12 original questions, 79 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

Find dydx for each of the following.

  1. y = (4x3 + 1)5
  2. y = ln(5x2 + 2)
  3. y = sec 4x
5 marks
Show answer
a) 60x2(4x3 + 1)4 b) 10x5x2 + 2 c) 4 sec 4x tan 4x
2
  1. Given y = x3e2x, show that dydx = x2e2x(a + bx), where a and b are integers to be found.
  2. Given y = 3x − 1x2 + 2, find dydx, giving your answer as a single fraction in its simplest form.
5 marks
Show answer
a) a = 3, b = 2 b) -3x2 + 2x + 6(x2 + 2)2
3
  1. The curve with equation y = 32x passes through the point P where x = 1. Find the exact gradient of the curve at P, giving your answer in the form k ln 3
  2. Given that x = cos y, where 0 < y < π, show that
    dydx = −1√(1 − x2)
6 marks
Show answer
a) 18 ln 3 b) shown
4

Given that x is measured in radians, prove, from first principles, that the derivative of sin 3x is 3 cos 3x

You may assume the formula for sin(A ± B) and that, as h → 0, sin hh → 1 and cos h − 1h → 0

5 marks
Show answer
the gradient of the chord tends to 3 cos 3x as h → 0
5

The curve C has equation y = (x2 − 3)ex

  1. Find the exact coordinates of the stationary points of C.
  2. Use d2ydx2 to determine the nature of each stationary point.
7 marks
Show answer
a) (1, -2e) and (-3, 6e-3) b) (1, -2e) is a minimum; (-3, 6e-3) is a maximum
6

The curve C has equation

x2 + 3xy − y2 = 27

  1. Find dydx in terms of x and y.
  2. The point P(3, 3) lies on C. Find an equation of the normal to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.
6 marks
Show answer
a) dydx = -2x + 3y3x - 2y b) x - 5y + 12 = 0
7

The curve C has equation

y = sin x1 + cos x,   0 < x < π

  1. Show that dydx = 11 + cos x
  2. Find the exact x coordinate of the point on C where the gradient is 23
5 marks
Show answer
a) shown b) x = π3
8

f(x) = x4 − 6x3 + 12x2 + 5

  1. Find f″(x)
  2. Find the set of values of x for which f(x) is concave.
  3. Find the coordinates of the points of inflection on the curve y = f(x), and show that neither of them is a stationary point.
6 marks
Show answer
a) f″(x) = 12x2 - 36x + 24 b) {x : 1 < x < 2} c) (1, 12) and (2, 21); f′(1) = 10, f′(2) = 8
9

Water is poured into a container in the shape of an inverted right circular cone at a constant rate of 40 cm3 s−1. When the depth of water is h cm, the radius of the water surface is 12h cm.

  1. Show that the volume of water, V cm3, is given by V = πh312
  2. Find the rate at which the depth of the water is increasing at the instant when h = 8, giving your answer to 3 significant figures.
  3. Find the exact rate at which the area of the water surface is increasing at this instant.
7 marks
Show answer
a) shown b) 0.796 cm s-1 c) 10 cm2 s-1
10

The curve C has equation

x2 + xy + y2 = 12

  1. Find dydx in terms of x and y.
  2. Find the coordinates of the points on C where the tangent is parallel to the x-axis.
  3. Find the coordinates of the points on C where the tangent is parallel to the y-axis.
8 marks
Show answer
a) dydx = -2x + yx + 2y b) (2, -4) and (-2, 4) c) (4, -2) and (-4, 2)
11

The number of people, N thousand, who have downloaded a new revision app t weeks after its launch is modelled by

N = 401 + 7e−0.5t,   t ≥ 0

  1. State the number of downloads at launch and the long-term limit of the number of downloads according to the model.
  2. Show that dNdt = N(40 − N)80
  3. Hence find the greatest rate of downloads, in thousands per week, according to the model, and the value of t at which it occurs. Give t to 3 significant figures.
  4. Explain why the graph of N against t has a point of inflection at this value of t.
9 marks
Show answer
a) 5000 at launch; limit 40 000 b) shown c) 5 thousand per week at t = 3.89 d) dN/dt has a maximum there: it increases before and decreases after, so the curve changes from convex to concave
12

The curve C has equation y = f(x), where

f(x) = ln xx2,   x > 0

  1. Show that f′(x) = 1 − 2 ln xx3
  2. Find the exact coordinates of the stationary point of C, and use f″(x) to show that it is a maximum.
  3. Find the set of values of x for which C is convex, giving the boundary in the form ek
  4. Given that f(x) → 0 as x → ∞, state, with a reason, the number of solutions of f(x) = k for 0 < k < 12e
10 marks
Show answer
a) shown b) (√e, 12e), a maximum c) {x : x > e5/6} d) two solutions

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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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