Edexcel International A Level Maths: Pure 4

Topic Questions

4.2 Implicit Differentiation

1
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8 marks

Find an expression for  fraction numerator d y over denominator d x end fraction when

(i)
x squared plus y equals 3 comma
(ii)
5 x to the power of 4 plus y squared minus 4 equals 0 comma
(iii)
sin space 3 x minus 3 y equals 0
(iv)
e to the power of x plus e to the power of y equals 2 x.

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2
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3 marks

Find the gradient of the curve with equation  3 y squared minus 2 x cubed equals 10  at the point (1 , 2).

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3
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3 marks

Show that if space x minus sin space y equals 0 space  then  fraction numerator d y over denominator d x end fraction space equals sec space y space.

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4a
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2 marks

The curve C has equation

y squared minus 4 x plus 2 equals 0.

Show that C intersects the x-axis at the point open parentheses 1 half space comma space 0 close parentheses.

4b
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3 marks
(i)
Find an expression for  fraction numerator d y over denominator d x end fraction.
(ii)
Explain why the curve C does not have any stationary points.

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5a
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2 marks

Show that the point open parentheses negative 4 over pi comma pi over 2 close parentheses lies on the curve with equation  2 cos space 2 y equals x y .

5b
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction.

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6a
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2 marks

Find the gradient of the curve with equation  space 12 x squared minus 4 y squared plus 24 equals 0 space spaceat the point (1 , 3).

6b
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2 marks

Hence, find an equation of the tangent to the curve at the point (1 , 3).

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7a
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3 marks

Given  space 3 x squared minus 2 y equals x y comma spacefind an expression for  fraction numerator d y over denominator d x end fraction.

7b
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2 marks

Hence show that the stationary points of space 3 x squared minus 2 y equals x y  lie on the line y equals 6 x.

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8a
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4 marks

The curve C has equation  x cubed plus 9 x y squared equals 54.

Find the gradient of the tangent to C at the point (3 , 1).

8b
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3 marks

Hence find the equation of the normal to C at (3 , 1), 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.

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1
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4 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of and y for the following

(i)
2 y e to the power of x plus 5 x squared y squared equals 8 comma
(ii)
3 x tan space y space equals 2 x squared.

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2a
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1 mark

Given that

y squared plus 4 x squared minus e to the power of y equals 0 comma

find the positive value of x when y equals 0.

2b
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4 marks

Hence, or otherwise, find the value of the gradient of

y squared plus 4 x squared minus e to the power of y equals 0

at the point where y equals 0 and x is positive.

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3
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3 marks

Show that if y equals arccos space 2 x , then

fraction numerator d y over denominator d x end fraction equals negative fraction numerator 2 over denominator square root of 1 minus 4 x squared end root end fraction.

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4a
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5 marks

The curve C has equation 2 x y squared minus x squared equals 16, Line L has equation x equals 4.

Show that the two points where C intersects L have equal gradients.

4b
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1 mark

What else can you deduce about the two points where C and L intercept?

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5
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5 marks

Verify that the point (-1 , 0) lies on the curve with equation

3 x e to the power of y plus 2 x plus 5 equals 4 y

and find the equation of the tangent to the curve at the point (-1 , 0).
Give your answer in the form a x plus b y plus c equals 0, where a, b and c are integers to be found.

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6a
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5 marks

Show that the derivative function of the curve given by 

ln space y minus 2 x y cubed equals 8

is given by

fraction numerator d y over denominator d x end fraction equals fraction numerator 2 y to the power of 4 over denominator 1 minus 6 x y cubed end fraction.

6b
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3 marks

Find the equation of the normal to the curve given in part (a) at the point where y equals 1, giving your answer in the form a x plus b y plus c equals 0 comma where a comma space b and c are integers to be found.

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7
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6 marks

Show that the stationary points on the curve with equation

x y squared minus 4 x squared equals 64

occur when x equals 4, and find the exact y-coordinates of the stationary points.

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8a
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1 mark

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

ln open parentheses space x y close parentheses space plus x y squared equals 1.

8b
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8 marks

The tangent at point A intercepts the x-axis at point B and the y-axis at point C.
Find the area of the triangle OBC.

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9
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3 marks

Show that

fraction numerator straight d over denominator straight d x end fraction open square brackets a to the power of k x end exponent close square brackets equals k a to the power of k x end exponent ln space a

where a and k are constants.

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1
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4 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of x and y for the following

(i)
2 x y plus y squared equals 4
(ii)
3 sin space y space minus y equals 2 x minus 1

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2
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4 marks

Find the gradient of the curve with equation

3 x squared y plus 4 x minus y equals 39

at the point with coordinates (2 , 3).

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3a
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2 marks

The curve C has equation

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

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

3b
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2 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of x and y.

3c
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2 marks

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

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4
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3 marks

Show that ifspace y equals arcsin space x space, then 

fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction

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5a
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2 marks

Show that the point (0 , π) lies on the curve with equation

3 tan space y equals 2 x y

5b
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction.

5c
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2 marks

Find the gradient at the point (0, π).

5d
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3 marks

Hence find an equation of the tangent to the curve at the point (0 ,π).

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6a
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4 marks

The curve C is defined by the equation ln space y equals 1 minus x y .

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

Show that

fraction numerator d y over denominator d x end fraction equals negative fraction numerator y squared over denominator 1 plus x y end fraction

6b
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3 marks

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

6c
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2 marks

Find the equation of the normal to C at point 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.

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7a
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of x and y given that

2 x squared minus y equals x y squared

7b
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2 marks

Show that fraction numerator d y over denominator d x end fraction equals 0 whenspace 4 x equals y squared.

7c
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3 marks

Hence, or otherwise, find the coordinates of the stationary points.

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8a
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2 marks

The curve C is given by the equation  e to the power of x y end exponent equals y minus x.

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

8b
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction

8c
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4 marks

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

y equals 2 x plus 1 and  2 y equals x plus 1

8d
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4 marks

The tangents meet at point Q.
Find the distance OQ, where O is the origin.

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9
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3 marks

Use implicit differentiation to show that

straight d over dx open square brackets a to the power of x close square brackets equals a to the power of x ln space a

where a is a constant.

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1
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5 marks

Find an expression for fraction numerator d y over denominator d x end fractionin terms of x and y for the following

(i)
e to the power of x y end exponent plus ln space open parentheses x y close parentheses equals cos e c space open parentheses x space close parentheses plus 4

(ii)
4 cos space open parentheses x squared y close parentheses minus 3 e to the power of x squared y end exponent equals 4 e to the power of y

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2
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5 marks

Find the gradient at the point wherespace x equals negative 2 space andspace y space is an integer on the curve with equation  x squared y squared minus 5 x equals 22 y.

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3
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3 marks

Show that if 2 y equals arctan space x squared, then

fraction numerator d y over denominator d x end fraction equals fraction numerator x over denominator 1 plus x to the power of 4 end fraction

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4
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4 marks

An ellipse has equation

x squared over 4 plus y squared over 9 equals 1

Find an expression for fraction numerator d y over denominator d x end fraction and hence show that the gradient of the ellipse at any point where it meets a line of the formspace y equals k x space space left parenthesis k not equal to 0 right parenthesis space is independent of x and y.

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5
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5 marks

The curve C is described by the equation

ln space y space plus x squared y squared equals 9.

Show that the tangents of the two points on C wherespace y equals 1 spacemeet at the point open parentheses 0 space comma 37 over 19 close parentheses.

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6a
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4 marks

The curve C is described by the equation

3 x squared plus 2 x y cubed plus 16 equals 0.

Show that the normal to C at the point where x equals negative 4 spaceis parallel to the normal to C at the point where x equals 4.

6b
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4 marks

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

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7
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8 marks

Find the stationary points and determine their nature for the curve with equation y squared equals 3 x squared minus 2 x y plus 3.

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8
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8 marks

The curve C is defined by

e to the power of sin space x y end exponent space equals 1 space space space space space space space space space space space space space space open curly brackets space y greater than 0 close curly brackets

Points A and B have coordinates open parentheses pi over 2 space comma space 2 close parentheses and open parentheses negative pi over 2 space comma space 2 close parentheses respectively.

The tangents to C at points A and B intersect at the point P.
The tangent to C at point A intersects the x-axis at point Q.
The tangent to C at point B intersects the x-axis at point R.

Find the area of triangle PQR.

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9
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3 marks

Show that

fraction numerator straight d over denominator straight d x end fraction open square brackets a to the power of x to the power of k end exponent close square brackets equals k a to the power of x to the power of k end exponent x to the power of k minus 1 end exponent ln space a

where a and k are constants.

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