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A LevelMaths: Pure 1CIEExam Questions5. Differentiation5.1 Differentiation

Differentiation (CIE A Level Maths: Pure 1)

Exam Questions

3 hours40 questions
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1
Sme Calculator3 marks

Differentiate

(i)
5 x comma

(ii)
2 x cubed

(iii)
x to the power of begin inline style 1 half end style end exponent.

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2a
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By first defining an appropriate function u, where u space equals space u open parentheses x close parentheses is a function of x,  show that the function ydefined by

      y equals left parenthesis x minus 7 right parenthesis cubed

may be written in the form

       y equals u to the power of n

where n is a positive integer.

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2b
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Using your result from part (a) along with the chain rule

      fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction

find fraction numerator d y over denominator d x end fraction for the function ydefined in part (a).  Be sure to give your answer entirely in terms of x.

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3a
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Write down the gradient of the line with equation y equals k, where k is a constant.

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3b
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Find the gradient at the point where x equals 8 for the following functions

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

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

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

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4a
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Given that  y equals left parenthesis x minus 3 right parenthesis to the power of 4 comma

By first defining an appropriate function u and writing y in the form y space equals space u to the power of n where n is a positive integer, use the chain rule to find fraction numerator d y over denominator d x end fraction in terms of x.

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4b
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(i)
By solving the equationfraction numerator d y over denominator d x end fraction space equals space 0, find the coordinates of any stationary points.

 

(ii)
By considering the value of fraction numerator d y over denominator d x end fraction on either side of any stationary points,

        determine the nature of those stationary points.

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4c
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Sketch the curve of y space equals space open parentheses x minus 3 close parentheses to the power of 4 .

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5
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(i)
Expand left parenthesis x plus 3 right parenthesis left parenthesis x minus 2 right parenthesis.

(ii)
Hence differentiatespace left parenthesis x plus 3 right parenthesis left parenthesis x minus 2 right parenthesis.

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6
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Given that  y equals 2 x to the power of begin inline style 1 half end style end exponent plus 3 x to the power of negative 1 end exponent, find  fraction numerator straight d y over denominator straight d x end fraction.

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7
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Find the x-coordinate of the point on the curve space y equals 5 x squared minus 16 x  where the gradient is 4.

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8
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Find the coordinates of the points on the curve  y equals 2 x cubed minus 9 x squared plus 12 x  where the gradient is 0.

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9
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Find  fraction numerator straight d y over denominator straight d x end fraction  when  y equals open parentheses square root of x close parentheses cubed plus fraction numerator space 2 over denominator square root of x end fraction.

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10a
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The function straight f left parenthesis x right parenthesis is given by

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

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

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10b
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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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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 equals 4 x squared minus 3 x plus 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 minus 3 x to the power of negative 1 end exponent

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

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3a
Sme Calculator2 marks

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

y equals left parenthesis 2 x plus 3 right parenthesis left parenthesis 3 x minus 1 right parenthesis

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3b
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y equals x cubed open parentheses 1 over x cubed minus 2 over x squared plus 3 over x close parentheses

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4a
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By first defining an appropriate function u, where u space equals space u open parentheses x close parentheses is a function of x,  show that the function y defined by

      y equals left parenthesis 3 x plus 11 right parenthesis to the power of 5

may be written as a function of u.

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4b
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Using your result from part (a) along with the chain rule

      fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction

find fraction numerator d y over denominator d x end fraction for the function y defined in part (a).  Be sure to give your answer entirely in terms of x.

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5a
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The function  is defined by text f end text left parenthesis x right parenthesis equals 2 x cubed minus x squared minus 4 x plus 3.

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

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5b
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Solve the equation straight f to the power of apostrophe open parentheses straight x close parentheses space equals space 0.

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6a
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A curve has the equation y space equals space 3 x minus space 4 x to the power of negative 2 end exponent comma space x space not equal to 0.

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

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6b
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Find the coordinates of the point on the curve where the gradient is 2.

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7a
Sme Calculator3 marks

Given that y equals left parenthesis 2 x plus 1 right parenthesis to the power of 4

use the chain rule to find fraction numerator d y over denominator d x end fraction

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7b
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find the coordinates of any stationary points and determine their nature

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7c
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sketch the curve.

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8a
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The function  is defined by straight f left parenthesis x right parenthesis equals x cubed minus 6 x squared minus c x plus 12.

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

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8b
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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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9a
Sme Calculator1 mark

A curve is described by the equation fraction numerator y over denominator left parenthesis x minus 3 right parenthesis end fraction equals x squared plus 1.

Make y the subject of the equation.

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9b
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Hence find fraction numerator d y over denominator d x end fraction.

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9c
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Find the coordinates of the point on the curve where the gradient is negative 2.

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10a
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The curve with equation y equals a x squared plus b x plus c has a gradient of negative 7 at the point open parentheses negative 1 comma 13 close parentheses, 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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10b
Sme Calculator1 mark

Hence find the values of a and b .

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10c
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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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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 equals negative 3 x cubed plus 5 x squared minus 3 x plus 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 minus 6 x to the power of negative 1 third end exponent

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2
Sme Calculator3 marks

Given that y equals fraction numerator 1 over denominator square root of x end fraction left parenthesis 1 plus 1 over x right parenthesis comma space space x greater than 0 comma space straight f i n d fraction numerator space d y over denominator d x space 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 equals left parenthesis 2 x minus 1 right parenthesis squared left parenthesis x plus 1 right parenthesis

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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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4a
Sme Calculator4 marks

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

y equals open parentheses 1 over x plus 2 x close parentheses to the power of 4

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4b
Sme Calculator4 marks

y equals fraction numerator 1 over denominator left parenthesis x cubed minus 1 right parenthesis squared end fraction

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5
Sme Calculator4 marks

The function straight f is defined by straight f open parentheses x space close parentheses space equals space straight x cubed minus 4 straight x squared plus 6 straight x minus 9. Show that there are no solutions to 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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6a
Sme Calculator3 marks

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

Show that fraction numerator d y over denominator d x end fraction equals a x to the power of negative 2 over 3 end exponent left parenthesis x plus b right parenthesis commawhere a and b are rational numbers to be found.

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6b
Sme Calculator2 marks

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

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7a
Sme Calculator3 marks

Given that  y equals left parenthesis x squared minus 2 x minus 3 right parenthesis to the power of 4

use the chain rule to find fraction numerator d y over denominator d x end fraction

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7b
Sme Calculator4 marks

find the coordinates of any stationary points and determine their nature

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7c
Sme Calculator3 marks

sketch the curve.

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8
Sme Calculator4 marks

A curve has the equation y equals 4 x cubed plus b x squared plus 3 x minus 17, where b 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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9
Sme Calculator2 marks

A curve is described by the equation 4 y squared minus 3 x to the power of 5 equals 0 comma space y greater than 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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10
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 left parenthesis negative 2 comma 0 right parenthesis comma space and a gradient of negative 10 at the point open parentheses 1 comma negative 3 close parentheses.  Find the values of a, b and c.

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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 equals negative 5 over 4 x cubed plus 3 over 5 x squared minus x square root of 2 plus pi

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1b
Sme Calculator2 marks

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 comma space x space greater than space 0 comma space straight f i n d space 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 equals fraction numerator 2 x cubed minus 5 x squared minus 3 x over denominator 2 x plus 1 end fraction

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3b
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y equals open parentheses square root of x plus 3 minus fraction numerator 4 over denominator square root of x end fraction close parentheses squared

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4a
Sme Calculator4 marks

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

      y equals open parentheses fraction numerator 1 over denominator square root of x end fraction plus 3 x squared close parentheses cubed

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4b
Sme Calculator4 marks

y equals square root of fraction numerator 1 over denominator square root of x plus fraction numerator 1 over denominator square root of x end fraction end fraction end root

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5
Sme Calculator5 marks

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

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6
Sme Calculator5 marks

The function straight f is defined by straight f open parentheses x close parentheses space equals space 2 straight x cubed plus px squared plus 3 straight x minus 16. Determine the range of values for p for which the equation straight f apostrophe space open parentheses straight x close parentheses space equals space 0has at least one real solution.

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7a
Sme Calculator3 marks

Given that y equals open parentheses 1 over x plus x close parentheses to the power of 4 comma space x not equal to 0

use the chain rule to find fraction numerator d y over denominator d x end fraction

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7b
Sme Calculator4 marks

find the coordinates of any stationary points and determine their nature

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7c
Sme Calculator3 marks

sketch the curve.

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8
Sme Calculator4 marks

The function straight f is defined by straight f open parentheses x close parentheses space equals space x to the power of straight n minus x comma space straight n element of straight natural numbers comma space n greater or equal than 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 x close parentheses space equals space 0. 

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9
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A curve is described by the equationfraction numerator square root of y over denominator negative 1 space plus space square root of x end fraction equals space 1 over x comma space space x greater than 1 Find fraction numerator d y over denominator d x end fraction.

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10
Sme Calculator5 marks

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

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