A LevelMathsEdexcelPureExam QuestionsDifferentiationImplicit Differentiation

Implicit Differentiation (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

4 hours41 questions
Easy
Medium
Hard
Very Hard
1a
2 marks

Find an expression for \frac{\text{d}y}{\text{d}x}, given that

x^2 + y = 3

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1b
2 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y, given that

5x^4 + y^2 - 4 = 0

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1c
2 marks

Find an expression for \frac{\text{d}y}{\text{d}x}, given that

\text{sin} \; 3x - 3y = 0

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1d
2 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y, given that

\text{e}^x + \text{e}^y = 2x

How did you do?

2
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3 marks

The curve C has equation

3y^2 - 2x^3 = 10

The point P(1, 2) lies on C.

Find the exact value of the gradient of C at the point P.

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3
3 marks

Given that

x - \text{sin} \; y = 0

show that

\frac{\text{d}y}{\text{d}x} = \text{sec} \; y

How did you do?

4a
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2 marks

The curve C has equation

y^2 - 4x + 2 = 0

Show that C intersects the x-axis at the point \left(\frac{1}{2}, 0\right).

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4b
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3 marks

(i) Find an expression for \frac{\text{d}y}{\text{d}x}.

(ii) Explain why the curve C does not have any stationary points.

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5a
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2 marks

The curve C has equation

2 \; \text{cos} \; 2y = xy

Show that the point P\left(-\frac{4}{\pi}, \frac{\pi}{2}\right) lies on C.

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5b
3 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y.

How did you do?

6a
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2 marks

The curve C has equation

12x^2 - 4y^2 + 24 = 0

The point P(1, 3) lies on C.

Find the gradient of C at the point P.

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6b
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2 marks

Hence, find an equation of the tangent to C at the point P.

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7a
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3 marks

The curve C has equation

3x^2 - 2y = xy

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y.

How did you do?

7b
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2 marks

Hence show that any stationary points on C lie on the line with equation y = 6x.

How did you do?

8a
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4 marks

The curve C has equation

x^3 + 9xy^2 = 54

The point P(3, 1) lies on C.

Find the gradient of the tangent to C at the point P.

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8b
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3 marks

Hence 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 to be found.

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1a
4 marks

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A curve has equation

x cubed plus 2 x y plus 3 y squared equals 47

Find fraction numerator straight d y over denominator straight d x end fraction in terms of x and y.

How did you do?

1b
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3 marks

The point P space open parentheses negative 2 comma space 5 close parentheses lies on the curve.

Find the equation of the normal to the curve at P, giving your answer in the form a x plus b y plus c equals 0, where a, b and c are integers to be found.

How did you do?

2
4 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y where appropriate, given that

(i) 2xy + y^2 = 4

(ii) 3 \; \text{sin} \; y - y = 2x - 1

How did you do?

3
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4 marks

The curve C has equation

3x^2 y + 4x - y = 41

The point P(2, 3) lies on C.

Find the exact value of the gradient of C at the point P.

How did you do?

4a
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2 marks

The curve C has equation

1 fifth x squared text e  end text to the power of y equals 5

Show that C intersects the x-axis at the points (-5, 0) and (5, 0).

How did you do?

4b
2 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y.

How did you do?

4c
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2 marks

Hence find the gradients of C at the two points where C intersects the x-axis.

How did you do?

5
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3 marks

Given that

y = \text{arcsin} \; x

show that

\frac{\text{d}y}{\text{d}x} = \frac{1}{\sqrt{1 - x^2}}

How did you do?

6a
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2 marks

The curve C has equation

3 \; \text{tan} \; y = 2xy

Show that the point P(0, \pi) lies on C.

How did you do?

6b
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3 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y.

How did you do?

6c
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2 marks

Find the exact value of the gradient of C at the point P.

How did you do?

6d
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3 marks

Hence find an equation of the tangent to C at the point P.

How did you do?

7a
4 marks

The curve C has equation

\text{ln} \; y = 1 - xy

The point P(1, 1) lies on C.

Show that

\frac{\text{d}y}{\text{d}x} = -\frac{y^2}{1 + xy}

How did you do?

7b
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3 marks

Find the gradient of the tangent to C at the point P, and hence find the gradient of the normal to C at P.

How did you do?

7c
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2 marks

Find an equation of the normal to C at the point P, giving your answer in the form ax + by + c = 0, where a, b and c are integers to be found.

How did you do?

8a
3 marks

The curve C has equation

2x^2 - y = xy^2

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y.

How did you do?

8b
2 marks

Show that \frac{\text{d}y}{\text{d}x} = 0 when 4x = y^2.

How did you do?

8c
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3 marks

Hence, or otherwise, find the exact coordinates of the stationary points on C.

How did you do?

9a
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2 marks

The curve C has equation

\text{e}^{xy} = y - x

Find the coordinates of the points where C crosses the coordinate axes.

How did you do?

9b
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3 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y.

How did you do?

9c
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4 marks

Show that the tangents to C at the points where it crosses the coordinate axes have equations

y = 2x + 1 and 2y = x + 1

How did you do?

9d
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4 marks

The two tangents meet at the point Q.

Find the exact distance OQ, where O is the origin.

How did you do?

10
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3 marks

Given that

y = a^x

where a is a positive constant, use implicit differentiation to show that

\frac{\text{d}y}{\text{d}x} = a^x \; \text{ln} \; a

How did you do?

1
5 marks
Graph showing a curve labelled 'C' in the first quadrant, starting at the origin and curving upwards to the right. Axes are labelled 'x' and 'y'.
Figure 8

Figure 8 shows a sketch of the curve C with equation y equals x to the power of x comma space space space x greater than 0.

Find, by firstly taking logarithms, the x coordinate of the turning point of C.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

How did you do?

2a
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4 marks

The curve C has equation

x squared space tan y equals 9 space space space space space space space space 0 less than y less than pi over 2

Show that

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator negative 18 x over denominator x to the power of 4 plus 81 end fraction

How did you do?

2b
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3 marks

Prove that C has a point of inflection at x equals fourth root of 27.

How did you do?

3
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4 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y, given that

(i) 2y \text{e}^x + 5x^2 y^2 = 8

(ii) 3x \; \text{tan} \; y = 2x^2

How did you do?

4a
4 marks

A curve has equation

2 x cubed plus y squared minus 3 x y equals 7

Show that

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 3 y minus 6 x squared over denominator 2 y minus 3 x end fraction

How did you do?

4b
2 marks

Find the equation of the normal to the curve at the point P space left parenthesis 2 comma 3 right parenthesis.

How did you do?

5a
1 mark

The curve C has equation

y^2 + 4x^2 - \text{e}^y = 0

Find the positive value of x when y = 0.

How did you do?

5b
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4 marks

Hence, or otherwise, find the value of the gradient of C at the point where y = 0 and x is positive.

How did you do?

6
3 marks

Given that

y = \text{arccos} \; 2x

show that

\frac{\text{d}y}{\text{d}x} = -\frac{2}{\sqrt{1 - 4x^2}}

How did you do?

7a
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5 marks

The curve C has equation

2xy^2 - x^2 = 16

The line l has equation x = 4.

Show that the gradient of C is the same at both points where C intersects l.

How did you do?

7b
1 mark

State what else can be deduced about these two points of intersection.

How did you do?

8
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5 marks

The curve C has equation

3x \text{e}^y + 2x + 5 = 4y

Verify that the point P(-1, 0) lies on C, and find an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0, where a, b and c are integers to be found.

How did you do?

9a
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5 marks

The curve C has equation

\text{ln} \; y - 2xy^3 = 8

Show that

\frac{\text{d}y}{\text{d}x} = \frac{2y^4}{1 - 6xy^3}

How did you do?

9b
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3 marks

Find an equation of the normal to C at the point where y = 1. Give your answer in the form ax + by + c = 0, where a, b and c are integers to be found.

How did you do?

10
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6 marks

The curve C has equation

xy^2 - 4x^2 = 64

Show that the stationary points on C occur when x = 4, and find the exact y-coordinates of these stationary points.

How did you do?

11a
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1 mark

A curve has equation

\text{ln}(xy) + xy^2 = 1

Verify that the point A(1, 1) lies on the curve.

How did you do?

11b
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8 marks

The tangent to the curve at the point A intersects the x-axis at the point B and the y-axis at the point C.

Find the exact area of the triangle OBC, where O is the origin.

How did you do?

12
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3 marks

Given that

y = a^{kx}

where a and k are constants with a > 0, use implicit differentiation to show that

\frac{\text{d}y}{\text{d}x} = k \, a^{kx} \; \text{ln} \; a

How did you do?

1a
4 marks

The curve C has equation

p x cubed plus q x y plus 3 y squared equals 26

where p and q are constants.

Show that

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator a p x squared plus b q y over denominator q x plus c y end fraction

where a, b and c are integers to be found.

How did you do?

1b
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5 marks

Given that

  • the point P open parentheses negative 1 comma space minus 4 close parentheses lies on C

  • the normal to C at P has equation 19 x plus 26 y plus 123 equals 0

find the value of p and the value of q.

How did you do?

2a
4 marks
Ellipse centred at O with axes North-South and East-West; points P and Q are the western-most and eastern-most extremities of the ellipse.
Figure 4

Figure 4 shows a sketch of the curve with equation x squared minus 2 x y plus 3 y squared equals 50.

Show that fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator y minus x over denominator 3 y minus x end fraction.

How did you do?

2b
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5 marks

The curve is used to model the shape of a cycle track with both x and y measured in km.

The points P and Q represent points that are furthest west and furthest east of the origin O, as shown in Figure 4.

Using part (a), find the exact coordinates of the point P.

How did you do?

2c
1 mark

Explain briefly how to find the coordinates of the point that is furthest north of the origin O. (You do not need to carry out this calculation).

How did you do?

3
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5 marks

Find an expression for \frac{\text{d}y}{\text{d}x} in terms of x and y, given that

(i) \text{e}^{xy} + \text{ln}(xy) = \text{cosec} \; x + 4

(ii) 4 \; \text{cos}(x^2 y) - 3 \text{e}^{x^2 y} = 4 \text{e}^y

How did you do?

4
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5 marks

The curve C has equation

x^2 y^2 - 5x = 22y

Find the exact value of the gradient of C at the point where x = -2 and y is an integer.

How did you do?

5
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3 marks

Given that

2y = \text{arctan}(x^2)

show that

\frac{\text{d}y}{\text{d}x} = \frac{x}{1 + x^4}

How did you do?

6
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4 marks

The curve C has equation

\frac{x^2}{4} + \frac{y^2}{9} = 1

Find an expression for \frac{\text{d}y}{\text{d}x} and hence show that the gradient of C at any point where it meets the line y = kx, where k is a non-zero constant, is independent of x and y.

How did you do?

7
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7 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation

\text{ln} \; y + x^2 y^2 = 9

Show that the tangents to C at the points where y = 1 intersect at the point \left(0, \frac{37}{19}\right).

How did you do?

8a
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4 marks

The curve C has equation

3x^2 + 2xy^3 + 16 = 0

Show that the normal to C at the point where x = -4 is parallel to the normal to C at the point where x = 4.

How did you do?

8b
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4 marks

Find the exact distance between the y-axis intercepts of these two normals.

How did you do?

9
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8 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation

y^2 = 3x^2 - 2xy + 3

Find the exact coordinates of the stationary points on C and determine their nature.

How did you do?

10
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8 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation

\text{e}^{\text{sin}(xy)} = 1 \quad \{y > 0\}

The points A\left(\frac{\pi}{2}, 2\right) and B\left(-\frac{\pi}{2}, 2\right) lie on C.

The tangent to C at A and the tangent to C at B intersect at the point P.

The tangent to C at A intersects the x-axis at the point Q.

The tangent to C at B intersects the x-axis at the point R.

Find the exact area of triangle PQR.

How did you do?

11
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3 marks

Given that

y = a^{x^k}

where a and k are constants with a > 0, use implicit differentiation to show that

\frac{\text{d}y}{\text{d}x} = k \, a^{x^k} x^{k-1} \; \text{ln} \; a

How did you do?