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

Differentiation: first principles and basic rules

Gradient as a limit, differentiating x^n including negative and fractional n, second derivatives.

12 original questions, 54 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

A curve has equation y = 4x5 − 3x2 + 7x − 9

  1. Find dydx
  2. Find the gradient of the curve at the point where x = −1
3 marks
Show answer
a) dydx = 20x4 - 6x + 7 b) 33
2

f(x) = 5x2 + 8√x − 3,   x > 0

Find f′(x), giving each term in its simplest form.

3 marks
Show answer
f'(x) = -10x3 + 4√x
3

y = 2x3 − 8x32,   x > 0

Find

  1. dydx
  2. d2ydx2
3 marks
Show answer
a) dydx = 6x2 - 12x12 b) d2ydx2 = 12x - 6√x
4

f(x) = 2x2 + 3x

Prove, from first principles, that f′(x) = 4x + 3

4 marks
Show answer
f'(x) = 4x + 3, shown from the limit of the chord gradient
5

Prove, from first principles, that the derivative of x3 is 3x2

4 marks
Show answer
The derivative of x3 is 3x2, shown from the limit of the chord gradient
6

The curve C has equation y = 3x2 − x

The point P(1, 2) lies on C. The point Q also lies on C and has x coordinate 1 + h, where h ≠ 0

  1. Show that the gradient of the chord PQ is 5 + 3h
  2. Explain how your answer to part (a) gives the gradient of the tangent to C at P, and state its value.
4 marks
Show answer
a) 5 + 3h (shown) b) as h → 0, Q → P and the chord gradient tends to 5
7

y = x2 + 62√x,   x > 0

Show that dydx = 3(x2 − 2)4x√x

4 marks
Show answer
dy/dx = 3(x2 - 2)/(4x√x), shown
8

A curve has equation y = (3 − √x)2,   x > 0

  1. Find dydx
  2. Find the value of x at the point on the curve where the gradient is 12
5 marks
Show answer
a) dydx = 1 - 3√x b) x = 36
9

f(x) = ax3 + bx2 − 3x, where a and b are constants.

Given that f′(2) = 17 and f″(2) = 22, find the value of a and the value of b.

5 marks
Show answer
a = 2, b = -1
10

The curve C has equation y = 16x + kx32,   x > 0, where k is a constant.

The gradient of C at the point where x = 4 is 5

  1. Find the value of k.
  2. Find the value of d2ydx2 at the point where x = 4
6 marks
Show answer
a) k = 2 b) 54
11

Water drains out of a tank. The depth of water, h cm, t minutes after the water starts to drain is modelled by

h = 64 − 16√t + t,   0 ≤ t ≤ 64

  1. Find dhdt
  2. Find the rate at which the depth is changing when t = 16, stating whether the depth is rising or falling.
  3. Find the time at which the depth is falling at 3 cm per minute.
6 marks
Show answer
a) dhdt = 1 - 8√t b) falling at 1 cm per minute c) t = 4 minutes
12

f(x) = ax2 + bx + c, where a, b and c are constants.

  1. Prove, from first principles, that f′(x) = 2ax + b

The curve with equation y = f(x)

  • passes through the point (0, 5)
  • has gradient 1 at the point where x = 1

and f″(x) = 6

  1. Find f(x).
7 marks
Show answer
a) f'(x) = 2ax + b (shown) b) f(x) = 3x2 - 5x + 5

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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