Differentiation of Polynomials: Fluency and Exam Drill - Worksheets, Questions and Revision

18 original exam-style questions - 4 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Calculus

C1D Differentiation of Polynomials: Fluency and Exam Drill

AQA 8365 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Differentiate y = x⁶ with respect to x.

(Total for Question 1 is 1 mark)

Differentiate y = x⁴ with respect to x.

(Total for Question 2 is 1 mark)

Differentiate y = 5x³ with respect to x.

(Total for Question 3 is 1 mark)

Differentiate y = 6x with respect to x.

(Total for Question 4 is 1 mark)

Differentiate y = 11 with respect to x.

(Total for Question 5 is 1 mark)

Differentiate y = 3x⁵ with respect to x.

(Total for Question 6 is 1 mark)

Find dy/dx for the curve y = 2x³ - 5x² + 4x - 1.

(Total for Question 7 is 2 marks)

Find dy/dx for the curve y = x⁴ + 3x³ - 2x.

(Total for Question 8 is 2 marks)

Expand the brackets, then find dy/dx for y = (x + 5)(x - 3).

(Total for Question 9 is 2 marks)

Expand the brackets, then find dy/dx for y = (3x - 2)(2x + 1).

(Total for Question 10 is 2 marks)

A curve has equation y = x³ - 5x + 2. Calculate the gradient of the curve at the point where x = 3.

(Total for Question 11 is 2 marks)

A curve has equation y = 2x² + 3x - 7. Calculate the gradient of the curve at the point where x = -2.

(Total for Question 12 is 2 marks)

A curve has equation y = x² - 10x + 3. Find the value of x for which the gradient of the curve is 4.

(Total for Question 13 is 3 marks)

The curve C has equation y = x³ - 5x² + 2x - 1. Show that the gradient of C at the point where x = 4 is 10.

(Total for Question 14 is 3 marks)

A curve has equation y = x² - 3x + 8. Find the equation of the tangent to the curve at the point where x = 5.

(Total for Question 15 is 4 marks)

A curve has equation y = x³ - 3x² - 24x + 5. Find the coordinates of the stationary points of the curve.

(Total for Question 16 is 4 marks)

A curve has equation y = x³ + 9x² + 28x - 3. Show that this curve is an increasing function for all values of x.

(Total for Question 17 is 4 marks)

A framer is making a rectangular picture frame from a single length of moulding 100 cm long, bent to form all four outer edges of the frame. Let x cm be the width of the frame and let A cm² be the area enclosed by the frame.

(a)Show that A = 50x - x².(2)

(b)Find the value of x that gives the maximum area, and calculate this maximum area.(2)

(Total for Question 18 is 4 marks)

Mark scheme · C1D Differentiation of Polynomials: Fluency and Exam Drill

Question 1

B1 dy/dx = 6x⁵ cao
Answer: dy/dx = 6x⁵

Question 2

B1 dy/dx = 4x³ cao
Answer: dy/dx = 4x³

Question 3

B1 dy/dx = 15x² cao
Answer: dy/dx = 15x²

Question 4

B1 dy/dx = 6 cao
Answer: dy/dx = 6

Question 5

B1 dy/dx = 0 cao
Answer: dy/dx = 0

Question 6

B1 dy/dx = 15x⁴ cao
Answer: dy/dx = 15x⁴

Question 7

M1 differentiates at least two terms correctly (e.g. 6x² and -10x seen)
A1 dy/dx = 6x² - 10x + 4 cao
Answer: dy/dx = 6x² - 10x + 4

Question 8

M1 differentiates at least two terms correctly (e.g. 4x³ and 9x² seen)
A1 dy/dx = 4x³ + 9x² - 2 cao
Answer: dy/dx = 4x³ + 9x² - 2

Question 9

M1 expands brackets to obtain y = x² + 2x - 15 oe
A1 dy/dx = 2x + 2 cao ft from expansion
Answer: dy/dx = 2x + 2

Question 10

M1 expands brackets to obtain y = 6x² - x - 2 oe
A1 dy/dx = 12x - 1 cao ft from expansion
Answer: dy/dx = 12x - 1

Question 11

M1 differentiates to obtain dy/dx = 3x² - 5 and substitutes x = 3
A1 gradient = 22 cao
Answer: gradient = 22

Question 12

M1 differentiates to obtain dy/dx = 4x + 3 and substitutes x = -2
A1 gradient = -5 cao
Answer: gradient = -5

Question 13

M1 differentiates and sets 2x - 10 = 4
M1 rearranges correctly to solve for x
A1 x = 7 cao
Answer: x = 7

Question 14

M1 differentiates correctly to obtain dy/dx = 3x² - 10x + 2
M1 substitutes x = 4 into dy/dx
A1 gradient = 10 fully shown cso (answer printed, all working correct)
Answer: 10 (shown)

Question 15

M1 finds the y-coordinate at x = 5 is 18
M1 differentiates to obtain dy/dx = 2x - 3 and finds the gradient at x = 5 is 7
M1 forms the tangent equation using y - 18 = 7(x - 5), dependent on both previous marks (dM1)
A1 y = 7x - 17 oe cao
Answer: y = 7x - 17

Question 16

M1 differentiates to obtain dy/dx = 3x² - 6x - 24
M1 sets dy/dx = 0 and factorises to (x - 4)(x + 2) = 0, dependent on previous M1 (dM1)
A1 x = 4 and x = -2 (both values) cao
A1 stationary points (4, -75) and (-2, 33) (both required, ft from the x-values)
Answer: Stationary points at (4, -75) and (-2, 33)

Question 17

M1 differentiates to obtain dy/dx = 3x² + 18x + 28
M1 attempts to complete the square on 3x² + 18x + 28
A1 dy/dx = 3(x + 3)² + 1 oe, correctly completed
A1 concludes that since (x+3)² ≥ 0 for all x, dy/dx ≥ 1 > 0 for all x, so the curve is increasing for all x (cso)
Answer: dy/dx = 3(x+3)² + 1, which is always at least 1, so the curve is increasing for all values of x (shown)

Question 18

(a) M1 uses half the perimeter to express the other side as (50 - x) and multiplies by x
(a) A1 A = 50x - x² fully shown cso
(a) Answer: A = 50x - x² (shown)
(b) M1 differentiates to obtain dA/dx = 50 - 2x, sets equal to 0 and solves, ft from part (a)
(b) A1 x = 25, giving a maximum area of 625 cm²
(b) Answer: x = 25 cm, maximum area = 625 cm²

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Differentiation of Polynomials: Fluency and Exam Drill - Worksheets, Questions and Revision

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