ASMaths: Pure 1CIEExam Questions5. Differentiation5.1 Differentiation

Differentiation (CIE AS Maths: Pure 1)

Exam Questions

3 hours40 questions
Easy
Medium
Hard
Very Hard
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.

How did you do?

Did this page help you?

2a
Sme Calculator2 marks

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.

How did you do?

2b
Sme Calculator3 marks

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.

How did you do?

Did this page help you?

3a
Sme Calculator1 mark

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

How did you do?

3b
Sme Calculator3 marks

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.

How did you do?

Did this page help you?

4a
Sme Calculator3 marks

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.

How did you do?

4b
Sme Calculator3 marks
(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.

How did you do?

4c
Sme Calculator3 marks

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

How did you do?

Did this page help you?

5
Sme Calculator3 marks
(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.

How did you do?

Did this page help you?

6
Sme Calculator2 marks

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.

How did you do?

Did this page help you?

7
Sme Calculator3 marks

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

How did you do?

Did this page help you?

8
Sme Calculator4 marks

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.

How did you do?

Did this page help you?

9
Sme Calculator3 marks

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.

How did you do?

Did this page help you?

10a
Sme Calculator3 marks

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.

How did you do?

10b
Sme Calculator3 marks

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

How did you do?

Did this page help you?

1a
Sme Calculator1 mark

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

How did you do?

1b
Sme Calculator2 marks

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

How did you do?

1c
Sme Calculator2 marks

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

How did you do?

Did this page help you?

2
Sme Calculator3 marks

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.

How did you do?

Did this page help you?

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

How did you do?

3b
Sme Calculator2 marks

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

How did you do?

Did this page help you?

4a
Sme Calculator2 marks

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.

How did you do?

4b
Sme Calculator3 marks

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.

How did you do?

Did this page help you?

5a
Sme Calculator2 marks

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.

How did you do?

5b
Sme Calculator2 marks

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

How did you do?

Did this page help you?

6a
Sme Calculator2 marks

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.

How did you do?

6b
Sme Calculator2 marks

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

How did you do?

Did this page help you?

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

How did you do?

7b
Sme Calculator3 marks

find the coordinates of any stationary points and determine their nature

How did you do?

7c
Sme Calculator3 marks

sketch the curve.

How did you do?

Did this page help you?

8a
Sme Calculator2 marks

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.

How did you do?

8b
Sme Calculator2 marks

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.

How did you do?

Did this page help you?

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.

How did you do?

9b
Sme Calculator1 mark

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

How did you do?

9c
Sme Calculator2 marks

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

How did you do?

Did this page help you?

10a
Sme Calculator2 marks

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.

How did you do?

10b
Sme Calculator1 mark

Hence find the values of a and b .

How did you do?

10c
Sme Calculator2 marks

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

How did you do?

Did this page help you?

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 3 x cubed plus 5 x squared minus 3 x plus square root of 13

How did you do?

1b
Sme Calculator2 marks

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

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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 left parenthesis 2 x minus 1 right parenthesis squared left parenthesis x plus 1 right parenthesis

How did you do?

3b
Sme Calculator3 marks

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

How did you do?

Did this page help you?

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

How did you do?

4b
Sme Calculator4 marks

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

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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.

How did you do?

6b
Sme Calculator2 marks

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

How did you do?

Did this page help you?

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

How did you do?

7b
Sme Calculator4 marks

find the coordinates of any stationary points and determine their nature

How did you do?

7c
Sme Calculator3 marks

sketch the curve.

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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

How did you do?

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

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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

How did you do?

3b
Sme Calculator4 marks

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

How did you do?

Did this page help you?

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

How did you do?

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

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?

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

How did you do?

7b
Sme Calculator4 marks

find the coordinates of any stationary points and determine their nature

How did you do?

7c
Sme Calculator3 marks

sketch the curve.

How did you do?

Did this page help you?

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. 

How did you do?

Did this page help you?

9
Sme Calculator3 marks

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.

How did you do?

Did this page help you?

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.

How did you do?

Did this page help you?