Edexcel A Level Maths: Pure

Topic Questions

A LevelMaths: PureEdexcelTopic Questions7. Differentiation7.1 Differentiation

7.1 Differentiation

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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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2
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Write down the formula that should be used as a starting point when explaining differentiation from first principles.

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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 1 third end exponent.

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4
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A student is trying to show that the derivative of 7 x squared is 14 x using first principles.
Their working is shown below.
Find and explain their error.

STEP 1

straight f open parentheses x close parentheses equals 7 x squared

 

STEP 2

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator straight f open parentheses x plus h close parentheses minus straight f left parenthesis x right parenthesis over denominator h end fraction

 

STEP 3

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator 7 open parentheses x plus h close parentheses squared minus 7 x squared over denominator h end fraction

 

STEP 4

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator 7 x squared plus 14 h x plus 7 h squared minus 7 x squared over denominator h end fraction

 

STEP 5

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of space fraction numerator h left parenthesis 14 x plus 7 h right parenthesis over denominator h end fraction

 

STEP 6

straight f apostrophe open parentheses x close parentheses equals 14 x plus 7 h

When  h=0,  14 x plus 7 h equals 14 x

therefore straight f apostrophe open parentheses x close parentheses equals 14 x

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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 apostrophe left parenthesis x right parenthesis.

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11
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Prove, from first principles, that the derivative of 4x is 4.

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1
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Prove, from first principles, that the derivative ofspace a x to the power of 2 space end exponent is 2ax, where a is a constant.

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2
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Prove, from first principles, that the derivative of space 2 x cubed spaceis 6 x squared.

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

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3b
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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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4
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Given that y equals fraction numerator space 1 over denominator square root of x end fraction open parentheses space 1 plus 1 over x close parentheses comma  x greater than 0, find fraction numerator d y over denominator d x end fraction.

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5a
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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 open parentheses 2 x minus 1 close parentheses to the power of 2 space end exponent open parentheses x plus 1 close parentheses

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

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

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7a
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A curve has the equation y equals 3 over 8 x 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 x over denominator d y end fraction space equals a x to the power of negative 2 over 3 end exponent open parentheses x plus b close parentheses, where a and b are rational numbers to be found.

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7b
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Hence find the coordinates of the point on the curve where the gradient is 0.

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8
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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
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A curve is described by the equation  4 y squared minus 3 x to the power of 5 equals 0 comma space space 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
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The curve with equationspace y equals a x squared plus b x plus c space has a gradient of 8 at the point open parentheses negative 2 comma space 0 close parentheses, and a gradient of -10 at the point open parentheses 1 comma space minus 3 close parentheses. Find the values of space a comma space b spaceand c.

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1
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Prove, from first principles, that the derivative of -3x is -3.

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2
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Prove, from first principles, that the derivative of 2x2 is 4x.

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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 4 x squared minus 3 x plus 19

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3b
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y equals x cubed minus 5 x squared plus 14 x minus 1

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

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4
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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 , space x greater than 0, find fraction numerator d y over denominator d x end fraction .

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5a
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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 open parentheses 2 x plus 3 close parentheses open parentheses 3 x minus 1 close parentheses

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5b
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y equals x cubed space left parenthesis 1 over x cubed minus 2 over x squared plus 3 over x right parenthesis

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6a
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The function f is defined by straight f open parentheses x close parentheses equals 2 x cubed minus x squared minus 4 x plus 3.

Find straight f apostrophe open parentheses x close parentheses.

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6b
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Solve the equation straight f apostrophe open parentheses x close parentheses equals 0.

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7a
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A curve has the equation y equals 3 x minus 4 x to the power of negative 2 end exponent comma space space x not equal to 0.

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

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7b
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Find the coordinates of the point on the curve where the gradient is 2.

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8a
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The function f is defined by straight f open parentheses x close parentheses equals x cubed minus 6 x squared minus c x plus 12.

Find straight f apostrophe open parentheses x close parentheses.

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8b
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Given that the equationspace straight f apostrophe open parentheses x close parentheses equals 0 space has exactly one real solution, find the value of c.

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9a
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A curve is described by the equation fraction numerator y over denominator x minus 3 end fraction space 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 -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 -7 at the point open parentheses negative 1 comma space 13 close parentheses, and a gradient of -3 at the point (1, 3).

By considering fraction numerator d y over denominator d x end fraction show that 2 a plus b equals negative 3 and negative 2 a plus b equals negative 7.

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10b
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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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1
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Prove, from first principles, that the derivative of  1 over x  is negative fraction numerator 1 over denominator space x squared space end fraction .

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2a
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Show that open parentheses square root of x plus h space end root minus square root of x close parentheses open parentheses square root of x plus h end root space plus square root of x close parentheses equals h.

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2b
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Prove, from first principles, that the derivative of square root of x is  fraction numerator 1 over denominator 2 square root of 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 equals negative 5 over 4 x cubed space plus space 3 over 5 x squared space minus space x square root of 2 space end root plus pi

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

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4
Sme Calculator4 marks

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

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5a
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 space fraction numerator 2 x cubed space minus space space 5 x squared space minus space space 3 x over denominator 2 x space plus space 1 end fraction

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5b
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y equals open parentheses space 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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6
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The function f is defined byspace straight f open parentheses x close parentheses equals 2 x cubed plus p x squared plus 3 x minus 16. Determine the range of values for p for which the equationspace straight f apostrophe open parentheses x close parentheses equals 0 space has at least one real solution.

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7
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A curve has the equation y equals x square root of x plus space fraction numerator 48 over denominator square root of x space end fraction ,  x greater than 0. Find the coordinates of the point on the curve where the gradient is 0.

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8
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The function f is defined by  straight f open parentheses x close parentheses equals x to the power of n minus x space comma space space 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 apostrophe open parentheses x close parentheses equals 0.

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9
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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 fraction numerator space 1 over denominator x end fraction space space comma space space space x greater than 1.  Find fraction numerator d y over denominator d x end fraction.

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10
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The curve with equation y equals a x squared plus b x plus c passes through the point (-1, 4).  At the point (2, 7) the gradient of the curve is 7.  Find the values of a, b and c.

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