A LevelMathsAQAPureExam QuestionsDifferentiationImplicit Differentiation

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

Exam code: 7357

3 hours36 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

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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

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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.

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7b
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2 marks

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

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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
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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.

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1b
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3 marks

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

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1c
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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

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1d
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4 marks

The two tangents meet at the point Q.

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

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2
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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?

3
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?

4
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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?

5a
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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?

5b
2 marks

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

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5c
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2 marks

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

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6
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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?

7a
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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?

7b
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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?

7c
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2 marks

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

How did you do?

7d
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3 marks

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

How did you do?

8a
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?

8b
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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.

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8c
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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.

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9a
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?

9b
2 marks

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

How did you do?

9c
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3 marks

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

How did you do?

1
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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?

2a
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?

2b
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?

3a
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?

3b
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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.

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4
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?

5a
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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.

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5b
1 mark

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

How did you do?

6
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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?

7a
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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?

7b
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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?

8
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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?

9a
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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?

9b
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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?

10
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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?

1
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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?

2
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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?

3
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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?

4
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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?

5
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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?

6a
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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?

6b
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4 marks

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

How did you do?

7
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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?

8
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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?

9
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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?