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Grade 8 · Higher · free GCSE maths worksheet

Differentiation

Edexcel IGCSE

Differentiating powers of x and finding the gradient of a curve at a point.

10 original questions, 32 marks. Type your answers below for an instant mark and a score, 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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How to do it

  1. Each term: multiply by the power, then take 1 off the power (x^n gives nx^(n-1))
  2. A plain x term becomes just its number; a constant on its own disappears
  3. Rewrite fractions as negative powers first: 3/x = 3x^-1, 4/x^2 = 4x^-2
  4. For the gradient at a point, substitute the x-value into dy/dx, not into y
  5. Tangent: gradient m from dy/dx, then y - y1 = m(x - x1)

Watch out: y = 3/x is 3x^-1, so dy/dx = -3x^-2 = -3/x^2: the power goes down to -2, it does not go to 0.

Worked example

The curve C has equation y = 2x3 − 5x2 + 4x − 7

  1. Find dydx
  2. Find the gradient of C at the point where x = 2

a) Multiply by the power, then take 1 off the power

dydx=6x2 - 10x + 4

The -7 on its own disappears

dydx = 6x2 - 10x + 4

b) Substitute x = 2 into dydx

gradient=6 × 22 - 10 × 2 + 4
=24 - 20 + 4

gradient = 8

The questions

1

y = x3 + 4x2 − 7x + 2

Find dydx

2 marks
Show answer
3x2 + 8x - 7
2

f(x) = 5x4 − 2x2 + 9

  1. Find f′(x)
  2. Work out the value of f′(−1)
3 marks
Show answer
a) 20x3 - 4x b) -16
3

The point P (4, −7) lies on the curve with equation y = x2 − 6x + 1

Find the gradient of the curve at P.

2 marks
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2
4

y = 6x + x2

  1. Find dydx
  2. Find the gradient of the curve at the point where x = 2
3 marks
Show answer
a) 2x - 6/x2 (accept 2x - 6x-2) b) 2.5
5

y = 4x2 + x

  1. Find dydx
  2. Find the gradient of the curve at the point where x = −1
3 marks
Show answer
a) 1 - 8/x3 (accept 1 - 8x-3) b) 9
6

A curve has equation y = x3 − 3x2 + 5x

Find the x-coordinates of the points on the curve where the gradient is 5.

3 marks
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x = 0 or x = 2
7

The curve with equation y = x2 − 8x + 3 has exactly one point where the gradient is zero.

Find the coordinates of this point.

3 marks
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(4, -13)
8

Water flows into a tank. The volume of water in the tank, V litres, t minutes after the water starts to flow is given by

V = 20t + 6t2 − t3   for 0 ≤ t ≤ 5

  1. Find dVdt
  2. Work out the value of dVdt when t = 3
  3. Explain, in context, what your answer to part (b) tells you.
4 marks
Show answer
a) 20 + 12t - 3t2 b) 29 c) the water is flowing into the tank at 29 litres per minute at t = 3
9

The curve C has equation y = x3 − 2x + 1

Find the equation of the tangent to C at the point where x = 2

Give your answer in the form y = mx + c

4 marks
Show answer
y = 10x - 15
10

The curve C has equation y = x(x − 3)2

  1. Find dydx
  2. Find the equation of the tangent to C at the point where x = 4
    Give your answer in the form y = mx + c
5 marks
Show answer
a) 3x2 - 12x + 9 b) y = 9x - 32

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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