International A LevelMaths: Pure 1EdexcelExam Questions4. Differentiation4.1 Differentiation

Differentiation (Edexcel International A Level Maths: Pure 1)

Exam Questions

3 hours43 questions
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Medium
Hard
Very Hard
1
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Differentiate

(i)
 5 x comma
(ii)
2 x cubed comma
(iii)
x to the power of 1 half end exponent.

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2a
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Write down the gradient of the line with equation y space equals space k, where k is a constant.

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2b
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Find the gradient at the point where x space equals space 8 for the following functions

(i)
straight f open parentheses straight x close parentheses space equals space 3 straight x squared comma

(ii)
straight f open parentheses straight x close parentheses space equals space 4 straight x cubed space minus space 2 straight x comma space

(iii)
straight f open parentheses straight x close parentheses space equals space 3 straight x to the power of 1 third end exponent.

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3
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(i)
Expand open parentheses x space plus space 3 close parentheses open parentheses space x space minus space 2 close parentheses.

(ii)
Hence differentiate open parentheses x space plus space 3 close parentheses space open parentheses x space minus space 2 close parentheses.

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4
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Given that y space equals space 2 x to the power of 1 half end exponent space plus space 3 x to the power of negative 1 end exponent, find fraction numerator d y over denominator d x end fraction.

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5
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Find the x-coordinate of the point on the curve y space equals space 5 x squared space minus space 16 x where the gradient is 4.

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6
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Find the coordinates of the points on the curve y space equals space 2 x to the power of 3 space end exponent minus space 9 x to the power of 2 space end exponent plus space 12 x where the gradient is 0.

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7
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Find fraction numerator d y over denominator d x end fraction when y space equals space open parentheses square root of x close parentheses cubed space plus space fraction numerator 2 over denominator square root of x end fraction.

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8a
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The function straight f open parentheses straight x close parentheses is given by

straight f open parentheses straight x close parentheses space equals space fraction numerator 2 straight x to the power of 1 third end exponent space plus space 3 straight x to the power of 2 over 3 end exponent over denominator straight x end fraction.

Show that straight f open parentheses straight x close parentheses can be written in the form straight f open parentheses x close parentheses space equals space a x to the power of b space plus space c x to the power of d, where a, b, c and d are constants to be found.

 

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8b
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Find straight f to the power of apostrophe space end exponent open parentheses straight x close parentheses.

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9a
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Find an expression for fraction numerator d y over denominator d x end fraction when y space equals space 3 x squared space minus space 2 x.

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9b
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Find the gradient of y space equals space 3 x squared space minus space 2 x at the points where

(i)
x space equals space 3 comma

(ii)
x space equals space minus 2.

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10
Sme Calculator5 marks
(i)
Find the gradient of the tangent at the point open parentheses 2 space comma space 3 close parentheses on the graph of y space equals space 2 x cubed space minus space 3 x to the power of 2 space end exponent minus space 1.

(ii)
Hence find the equation of the tangent at the point open parentheses 2 space comma space 3 close parentheses.

 

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11a
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For the graph with equation y space equals space 3 x space minus 1 half x squared, find the gradient of the tangent at the point where x space equals space 5.

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11b
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(i)
Find the gradient of the normal at the point where x space equals space 5.

(ii)
Hence find the equation of the normal at the point where x space equals space 5.

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1a
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For each of the following, find fraction numerator d y over denominator d x end fraction in terms of x:

y space equals space 4 x to the power of 2 space end exponent minus space 3 x space plus space 19

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1b
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y equals x cubed minus 5 x squared plus 14 x minus 1

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1c
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y equals 4 x to the power of 3 over 2 end exponent space minus space 3 x to the power of negative 1 end exponent

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2
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Given that y space equals space square root of x space end root space plus space fraction numerator 1 over denominator square root of x end fraction comma space x space greater than space 0 comma find fraction numerator d y over denominator d x end fraction.

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3a
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For each of the following, find fraction numerator d y over denominator d x end fraction in terms of x:

y space equals space open parentheses 2 x space plus space 3 close parentheses space open parentheses 3 x space minus space 1 close parentheses

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3b
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y space equals space x to the power of 3 space space end exponent open parentheses 1 over x cubed space minus space 2 over x squared space plus space 3 over x close parentheses

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4a
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The function f is defined by straight f space open parentheses straight x close parentheses space equals space 2 straight x to the power of 3 space end exponent minus space straight x squared space minus space 4 straight x space plus space 3.

Find straight f to the power of apostrophe space end exponent open parentheses x close parentheses.

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4b
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Solve the equation straight f to the power of apostrophe space end exponent open parentheses x close parentheses space equals space 0.

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5a
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A curve has the equation y space equals space 3 x space minus space 4 x to the power of negative 2 end exponent comma space x space not equal to space 0.

Find fraction numerator d y over denominator d x end fraction.

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5b
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Find the coordinates of the point on the curve where the gradient is 2.

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6a
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The function f is defined by straight f space open parentheses straight x close parentheses space equals space straight x cubed space minus space 6 straight x squared space minus space cx space plus space 12.

Find straight f to the power of apostrophe space end exponent open parentheses straight x close parentheses.

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6b
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Given that the equation straight f to the power of apostrophe space end exponent open parentheses straight x close parentheses space equals space 0 has exactly one real solution, find the value of c.

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7a
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A curve is described by the equation fraction numerator y over denominator x space minus space 3 end fraction space equals space x squared space plus 1. space   

Make y the subject of the equation.

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7b
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Hence find fraction numerator d y over denominator d x end fraction.

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7c
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Find the coordinates of the point on the curve where the gradient is negative space 2.

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8a
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The curve with equation y space equals space a x squared space plus space b x space plus space c has a gradient of negative 7 at the point open parentheses negative 1 comma space 13 close parentheses comma and a gradient of negative 3 at the point open parentheses 1 comma 3 close parentheses. 
By considering fraction numerator d y over denominator d x end fraction show that 2 a space plus space b space equals space minus 3 and negative 2 a space plus space b space equals space minus 7.

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8b
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Hence find the values of a and b.

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8c
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By considering a point that you know to be on the curve, find the value of c.

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9a
Sme Calculator1 mark

The curve C has equation y space equals space 2 x to the power of 3 space end exponent space minus space 3 x squared space plus space 4 x space minus 3.

 

Show that the point P space open parentheses 2 comma 9 close parentheses lies on C .

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9b
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Show that the value of fraction numerator d y over denominator d x end fraction  at P is 16.

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9c
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Find an equation of the tangent to C at P.

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10a
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The curve C has equation y space equals space 3 x squared space minus space 6 space plus space 4 over x. The point P space open parentheses 1 comma 1 close parentheses  lies on C.

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

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10b
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Show that an equation of the normal to C at point P is x space plus space 2 y space equals space 3.

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10c
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This normal cuts the -axis at the point Q .

Find the length of P Q, giving your answer as an exact value.

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11a
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Given that y space equals space 2 x cubed space minus space 8 square root of x, find

 fraction numerator d y over denominator d x end fraction

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11b
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fraction numerator d to the power of 2 space end exponent y over denominator d x squared end fraction

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1a
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For each of the following, find fraction numerator d y over denominator d x end fraction in terms of x:

 y space equals space minus 3 x cubed plus space 5 x squared space minus space 3 x space plus space square root of 13

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1b
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y equals 9 x to the power of 1 third end exponent space minus 6 x to the power of negative 1 third end exponent

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2
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Given that y space equals space fraction numerator 1 over denominator square root of x end fraction space open parentheses 1 space plus space 1 over x close parentheses space comma space x space greater than space 0 comma spacefind fraction numerator d y over denominator d x end fraction.

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3a
Sme Calculator3 marks

For each of the following, find fraction numerator d y over denominator d x end fraction in terms of x:

y space equals space open parentheses 2 x space minus space 1 close parentheses squared space open parentheses x space plus space 1 close parentheses

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3b
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y equals 1 over x to the power of 5 left parenthesis x squared plus square root of x minus 1 right parenthesis

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4
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The function straight f is defined by straight f space open parentheses straight x close parentheses space equals space straight x cubed space minus space 4 straight x squared space plus space 6 straight x space minus space 9. Show that there are no solutions to the equation straight f to the power of apostrophe space end exponent open parentheses straight x close parentheses space equals space 0.

 

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5a
Sme Calculator3 marks

A curve has the equation y space equals space 3 over 8 x to the power of 4 over 3 end exponent space minus space 12 x to the power of 1 third end exponent.

Show that fraction numerator d y over denominator d x end fraction space equals space a x to the power of negative 2 over 3 end exponent space open parentheses x space plus b close parentheses comma where a and b are rational numbers to be found.

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5b
Sme Calculator2 marks

Hence find the coordinates of the point on the curve where the gradient is 0.

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6
Sme Calculator4 marks

A curve has the equation y space equals space 4 x cubed space plus space b x squared space plus space 3 x space minus space 17 comma where  is a constant.  Given that there is only one point on the curve where the gradient is zero, determine the possible values of  b.

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7
Sme Calculator2 marks

A curve is described by the equation 4 y to the power of 2 space end exponent minus space 3 x to the power of 5 space equals space 0 comma space y space greater than space 0.

By rearranging the equation to make y the subject, find fraction numerator d y over denominator d x end fraction.

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8
Sme Calculator5 marks

The curve with equation y equals a x squared plus b x plus c has a gradient of 8 at the point open parentheses negative 2 space comma space 0 close parentheses, and a gradient of negative 10 at the point open parentheses 1 comma space minus 3 close parentheses.  Find the values of a comma space b and c.

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9
Sme Calculator5 marks

The curve C has equation y space equals space 3 x squared space minus space 6 x space plus space square root of 2 x. end root The point P space open parentheses 2 comma space 2 close parentheses  lies on C.

Find an equation of the tangent to C at P.

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10
Sme Calculator6 marks

The curve C has equation y space equals fraction numerator 9 over denominator square root of 3 x end root end fraction space minus space 3 over x. The point P space open parentheses 3 comma space 2 close parentheses lies on C.

The normal to C at P intersects the -axis at the point Q.

Find the coordinates of Q.

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11a
Sme Calculator3 marks

Given that y space equals space 4 over x space minus space cube root of 27 over x end root, find

fraction numerator d y over denominator d x end fraction

 

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11b
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fraction numerator d squared y over denominator d x squared end fraction

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1a
Sme Calculator2 marks

For each of the following, find fraction numerator d y over denominator d x end fraction  in terms of x :

y space equals space minus space 5 over 4 x cubed space plus space 3 over 5 x squared space space minus space x square root of 2 space plus space straight pi

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1b
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y equals 3 over 2 x to the power of 4 over 5 end exponent minus 10 over 3 x to the power of negative 4 over 5 end exponent

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2
Sme Calculator4 marks

Given that y equals open parentheses 1 over x minus fraction numerator 1 over denominator x square root of x end fraction close parentheses squared, x space greater than space 0 comma find fraction numerator d y over denominator d x end fraction.

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3a
Sme Calculator3 marks

For each of the following, find fraction numerator d y over denominator d x end fractionin terms of x:

y space equals space fraction numerator 2 x cubed space minus space 5 x to the power of 2 space end exponent minus space 3 x over denominator 2 x plus space 1 end fraction

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3b
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4
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5
Sme Calculator5 marks

A curve has the equation y space equals space x square root of x space end root plus space fraction numerator 48 over denominator square root of x end fraction comma space x space greater than space 0. Find the coordinates of the point on the curve where the gradient is 0.

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6
Sme Calculator4 marks

The function f is defined by straight f space open parentheses straight x close parentheses space equals space straight x to the power of straight n space end exponent minus space straight x space comma n space element of space straight natural numbers comma space n space greater or equal than space 2. Determine the relationship between the value of n and the number of real solutions to the equation straight f to the power of apostrophe space end exponent open parentheses straight x close parentheses space equals space 0.

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7
Sme Calculator3 marks

A curve is described by the equation fraction numerator square root of y over denominator negative 1 space plus space square root of x end fraction space equals space 1 over x comma space x space greater than space 1. Findfraction numerator d y over denominator d x end fraction. 

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8
Sme Calculator5 marks

The curve with equation  y equals a x squared plus b x plus c passes through the point open parentheses negative 1 comma space 4 close parentheses.  At the point open parentheses 2 comma 7 close parentheses the gradient of the curve is 7.  Find the values of a comma space b and c.

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9a
Sme Calculator7 marks

A curve has equation y equals 5 minus left parenthesis x minus 3 right parenthesis squared. 

A is the point on the curve with x coordinate 0, and B is the point on the curve with x coordinate 6. 

C is the point of intersection of the tangents to the curve at A and B.

Find the coordinates of point C.

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9b
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Calculate the area of triangle A B C.

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10a
Sme Calculator6 marks

A curve is described by the equation y space equals space straight f space open parentheses straight x close parentheses, where

straight f open parentheses space straight x close parentheses space equals space fraction numerator 1 over denominator square root of straight x end fraction comma space straight x space greater than space 0

P is the point on the curve such that the normal to the curve at P also passes through the origin.

Find the coordinates of point P. Give your answer in the form open parentheses 2 to the power of a space comma space 2 to the power of b close parentheses, where a and b are rational numbers to be found.

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10b
Sme Calculator1 mark

Write down the equation of the normal to the curve at P.

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10c
Sme Calculator4 marks

Show that an equation of the tangent to the curve at P is

open parentheses 2 to the power of 1 third end exponent close parentheses space x space plus open parentheses space 2 to the power of 5 over 6 end exponent close parentheses space y space equals space 3

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