ks5 full coverage: differentiation (yr2)
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KS5 "Full Coverage": Differentiation (Yr2)
This worksheet is designed to cover one question of each type seen in past papers, for each A
Level topic. This worksheet was automatically generated by the DrFrostMaths Homework
Platform: students can practice this set of questions interactively by going to
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teachers), and using the βRevisionβ tab.
Question 1 Categorisation: Use the chain rule (and differentiate natural logs).
[Edexcel C3 June 2011 Q1a] Differentiate with respect to π₯
ππ (π₯2 + 3π₯ + 5)
π( ππ (π₯2+3π₯+5))
ππ₯ ..........................
Question 2 Categorisation: Use the product rule (and differentiate ππ)
[Edexcel C3 June 2007 Q3a] A curve C has equation π¦ = π₯2ππ₯ .
Find ππ¦
ππ₯ , using the product rule for differentiation.
ππ¦
ππ₯= ..........................
Question 3 Categorisation: Use the chain rule combined with product rule.
[Edexcel C3 June 2009 Q4ia]Differentiate with respect to π₯
π₯2 πππ 3π₯
π(π₯2 πππ 3π₯)
ππ₯= ..........................
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Question 4 Categorisation: Use the quotient rule.
[Edexcel C3 June 2016 Q2a]
π¦ =4π₯
π₯2 + 5
Find ππ¦
ππ₯ , writing your answer as a single fraction in its simplest form.
ππ¦
ππ₯= ..........................
Question 5 Categorisation: As above.
[Edexcel C3 June 2013(R) Q5a]
Differentiate πππ 2π₯
βπ₯ with respect to π₯ .
..........................
Question 6 Categorisation: Further practice of product with chain rule.
[Edexcel A2 SAM P2 Q3]
Given π¦ = π₯(2π₯ + 1)4 , show that
ππ¦
ππ₯= (2π₯ + 1)π(π΄π₯ + π΅)
where π , π΄ and π΅ are constants to be found.
..........................
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Question 7 Categorisation: Differentiate trig functions raised to a power.
[Edexcel C3 June 2005 Q2ai]
Differentiate with respect to π₯
3 π ππ 2π₯ + π ππ 2π₯
π
ππ₯(3 π ππ 2π₯ + π ππ 2π₯) = ..........................
Question 8 Categorisation: As above, but with a multiple of π.
[Edexcel C3 June 2013(R) Q5b]
Show that π
ππ₯( π ππ 23π₯) can be written in the form
π( π‘ππ 3π₯ + π‘ππ 33π₯)
where π is a constant to be found.
π = ..........................
Question 9
Categorisation: Appreciate that ππ(ππ) differentiates to π
π regardless of π.
[Edexcel C3 June 2012 Q7ai]
Differentiate with respect to π₯ ,
π₯12 ππ (3π₯)
π(π₯12 ππ (3π₯))
ππ₯= ..........................
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Question 10
Categorisation: Appreciate that π
π(π)π(π) can be written as π(π)βππ(π)βπ before
differentiating.
[Edexcel C3 June 2013 Q5c]
ππ¦
ππ₯=
1
6π₯(π₯ β 1)12
Find an expression for π2π¦
ππ₯2 in terms of π₯ . Give your answer in its simplest form.
π2π¦
ππ₯2 = ..........................
Question 11 Categorisation: Differentiate in a modelling context.
[Edexcel C3 June 2006 Q4c]
A heated metal ball is dropped into a liquid. As the ball cools, its temperature, π, π‘ minutes
after it enters the liquid, is given by
π = 400πβ0.05π‘ + 25 , π‘ β₯ 0
Find the rate at which the temperature of the ball is decreasing at the instant when π‘ = 50
.......................... Β°πΆ /min
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Question 12 Categorisation: As above.
[Edexcel C3 June 2017 Q8b]
The number of rabbits on an island is modelled by the equation
π =100πβ0.1π‘
1+3πβ0.9π‘ + 40 , π‘ β β , π‘ β₯ 0
where π is the number of rabbits, π‘ years after they were introduced onto the island.
A sketch of the graph of π against π‘ is shown in Figure 3.
Find ππ
ππ‘
ππ
ππ‘= ..........................
Question 13
Categorisation: Use the fact that π π
π π= π Γ·
π π
π π
[Edexcel C3 June 2017 Q7ii]
Given π₯ = ππ ( π ππ 2π¦) , 0 < π¦ <π
4
find ππ¦
ππ₯ as a function of π₯ in its simplest form.
ππ¦
ππ₯= ..........................
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Question 14
Categorisation: Determine π π
π π for expressions of the form π = π(π) where π is a
trigonometric function. Note that the specification effectively expects students to
differentiate expressions such as ππππππ π, even though questions wouldnβt
explicitly ask as such.
[Edexcel C3 June 2013(R) Q5c]
Given π₯ = 2 π ππ (π¦
3) , find
ππ¦
ππ₯ in terms of π₯ , simplifying your answer.
ππ¦
ππ₯= ..........................
Question 15 Categorisation: As above.
[Edexcel C3 June 2016 Q5ii]
Given π₯ = π ππ 22π¦ , 0 < π¦ <π
4 , find
ππ¦
ππ₯ as a function of π¦ .
Write your answer in the form
ππ¦
ππ₯= π πππ ππ (ππ¦) , 0 < π¦ <
π
4 ,
where π and π are constants to be determined.
..........................
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Question 16 Categorisation: Find the range of values for which a function is increasing or
decreasing.
[Edexcel C3 June 2016 Q2b Edited]
π¦ =4π₯
π₯2 + 5
ππ¦
ππ₯=
20 β 4π₯2
(π₯2 + 5)2
Hence find the set of values of π₯ for which ππ¦
ππ₯< 0 .
..........................
Question 17 Categorisation: Use differentiation to determine a turning point.
[Edexcel C3 June 2016 Q5i]
Find, using calculus, the π₯ coordinate of the turning point of the curve with equation
π¦ = π3π₯ πππ 4π₯ , π
4β€ π₯ <
π
2
Give your answer to 4 decimal places.
..........................
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Question 18 Categorisation: Use differentiation in the context of numerical methods.
[Edexcel A2 Specimen Papers P2 Q6b Edited]
Figure 2 shows a sketch of the curve with equation π¦ = π(π₯) , where
π(π₯) = (8 β π₯) ππ π₯ , π₯ > 0
The curve cuts the π₯ -axis at the points π΄ and π΅ and has a maximum turning point at π , as
shown in Figure 2.
Show that the π₯ coordinate of π satisfies
π₯ =π
1 + ππ π₯
where π is a constant to be found.
..........................
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Question 19 Categorisation: Determine a turning point with a more difficult expression.
[Edexcel A2 SAM P1 Q15a Edited]
Figure 5 shows a sketch of the curve with equation π¦ = π(π₯) , where
π(π₯) =4 π ππ 2π₯
πβ2π₯β1 , 0 β€ π₯ β€ π
The curve has a maximum turning point at π and a minimum turning point at π as shown in
Figure 5.
Show that the π₯ coordinates of point π and point π are solutions of the equation π‘ππ 2π₯ =
βπ , where π is a constant to be found.
..........................
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Question 20 Categorisation: As above.
[Edexcel C3 June 2013 Q4a]
π(π₯) = 25π₯2π2π₯ β 16 , π₯ β β
One of the turning points of the curve with equation π¦ = π(π₯) is (0, β16) . Using calculus,
find the exact coordinates of the other turning point.
..........................
Question 21 Categorisation: Determine the equation of a tangent to the curve.
[Edexcel C3 June 2009 Q4ii]
A curve C has the equation
π¦ = β4π₯ + 1 , π₯ > β1
4 , π¦ > 0
The point P on the curve has π₯ -coordinate 2. Find an equation of the tangent to C at P
in the form ππ₯ + ππ¦ + π = 0 , where π , π and π are integers.
..........................
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Question 22 Categorisation: As above, but where π = π(π)
[Edexcel C3 June 2014 Q3b]
The curve πΆ has equation π₯ = 8π¦ π‘ππ 2π¦ .
The point π has coordinates (π,π
8) . Find the equation of the tangent to πΆ at π in the form
ππ¦ = π₯ + π , where the constants π and π are to be found in terms of π .
..........................
Question 23 Categorisation: Determine the equation of a normal to the curve.
[Edexcel C3 June 2010 Q2]
A curve C has equation
π¦ =3
(5 β 3π₯)2 , π₯ β
5
3
The point P on C has π₯ -coordinate 2.
Find an equation of the normal to C at P in the form ππ₯ + ππ¦ + π = 0 , where π , π and
π are integers.
..........................
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Question 24 Categorisation: As above.
[Edexcel C3 June 2011 Q7b Edited]
π(π₯) =5
(2π₯ + 1)(π₯ + 3)
π₯ β Β±3, π₯ β β1
2
The curve C has equation π¦ = π(π₯) . The point π (β1, β5
2) lies on C.
Find an equation of the normal to C at P.
..........................
Question 25 Categorisation: As above, but in the context of having done algebraic long division
first.
[Edexcel C3 June 2016 Q6b Edited]
π(π₯) =π₯4 + π₯3 β 3π₯2 + 7π₯ β 6
π₯2 + π₯ β 6, π₯ > 2, π₯ β β
Given that
π₯4 + π₯3 β 3π₯2 + 7π₯ β 6
π₯2 + π₯ β 6β‘ π₯2 + 3 +
4
π₯ β 2
Hence or otherwise, using calculus, find an equation of the normal to the curve with equation
π¦ = π(π₯) at the point where π₯ = 3.
..........................
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Question 26 Categorisation: Determine the gradient of parametric equations.
[Edexcel C4 June 2017 Q1a]
The curve C has parametric equations
π₯ = 3π‘ β 4 π¦ = 5 β6
π‘ π‘ > 0
Find ππ¦
ππ₯ in terms of π‘ .
ππ¦
ππ₯= ..........................
Question 27 Categorisation: As above, but with trigonometric expressions.
[Edexcel A2 SAM P1 Q13a]
The curve πΆ has parametric equationsπ₯ = 2 πππ π‘ , π¦ = β3 πππ 2π‘ , 0 β€ π‘ β€ π
Find an expression for ππ¦
ππ₯ in terms of π‘ .
ππ¦
ππ₯= ..........................
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Question 28 Categorisation: Determine a particular gradient at a point on a parametric curve.
[Edexcel C4 June 2013(R) Q7a]
Figure 2 shows a sketch of the curve C with parametric equations
π₯ = 27 π ππ 3π‘ , π¦ = 3 π‘ππ π‘ , 0 β€ π‘ β€π
3
Find the gradient of the curve C at the point where π‘ =π
6 .
..........................
Question 29 Categorisation: Determine the equation of a normal to a parametric curve.
[Edexcel C4 Jan 2011 Q6a]
The curve C has parametric equations
π₯ = ππ π‘ , π¦ = π‘2 β 2 , π‘ > 0 .
Find an equation of the normal to C at the point where π‘ = 3 .
..........................
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Question 30 Categorisation: As above, but involving exponential terms of the form ππ.
[Edexcel C4 Jan 2013 Q5c Edited]
Figure 2 shows a sketch of part of the curve C with parametric equations
π₯ = 1 β1
2π‘ , π¦ = 2π‘ β 1 .
The curve crosses the π¦ -axis at the point π΄ and crosses the π₯ -axis at the point π΅ .
The point π΄ has coordinates (0,3) .
Find an equation of the normal to C at the point π΄ .
..........................
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Question 31 Categorisation: Use implicit differentiation.
[Edexcel C4 June 2016 Q3a] The curve C has equation
2π₯2π¦ + 2π₯ + 4π¦ β πππ (ππ¦) = 17
Use implicit differentiation to find ππ¦
ππ₯ in terms of π₯ and π¦ .
ππ¦
ππ₯= ..........................
Question 32 Categorisation: Use implicit differentiation to find the gradient at a specific point.
[Edexcel C4 June 2017 Q4a]
The curve C has equation
4π₯2 β π¦3 β 4π₯π¦ + 2π¦ = 0
The point P with coordinates (β2, 4) lies on C.
Find the exact value of ππ¦
ππ₯ at the point P.
ππ¦
ππ₯= ..........................
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Question 33 Categorisation: As above, but involving an exponential term.
[Edexcel C4 June 2013(R) Q2]
The curve C has equation
3π₯β1 + π₯π¦ β π¦2 + 5 = 0
Show that ππ¦
ππ₯ at the point (1,3) on the curve C can be written in the form
1
π ππ (ππ3) ,
where π and π are integers to be found.
ππ¦
ππ₯= ..........................
Question 34 Categorisation: Determine a turning point on a parametric curve by solving
simultaneously with the original equation.
[Edexcel C4 June 2015 Q2b Edited]
The curve C has equation π₯2 β 3π₯π¦ β 4π¦2 + 64 = 0
The gradient function ππ¦
ππ₯ can be expressed as
ππ¦
ππ₯=
2π₯ β 3π¦
3π₯ + 8π¦
Find the coordinates of the points on C where ππ¦
ππ₯= 0 .
(Solutions based entirely on graphical or numerical methods are not acceptable.)
..........................
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Question 35 Categorisation: As above.
[Edexcel C4 June 2014(R) Q3b Edited]
π₯2 + π¦2 + 10π₯ + 2π¦ β 4π₯π¦ = 10
An expression for ππ¦
ππ₯ can be expressed as
ππ¦
ππ₯=
π₯ + 5 β 2π¦
2π₯ β π¦ β 1
Find the values of π¦ for which ππ¦
ππ₯= 0 .
..........................
Question 36 Categorisation: Connect different rates of change using the chain rule.
[Edexcel C4 June 2012 Q2b Edited]
Figure 1 shows a metal cube which is expanding uniformly as it is heated. At time π‘ seconds,
the length of each edge of the cube is π₯ cm, and the volume of the cube is π cm3.
Given that the volume, π cm3, increases at a constant rate of 0.048 cm3 sβ1, find ππ₯
ππ‘ when
π₯ = 8 .
ππ₯
ππ‘= ..........................
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Question 37 Categorisation: As above.
[Edexcel C4 June 2012 Q2c Edited]
(Continued from above) Given that the volume, π cm3, increases at a constant rate of
0.048 cm3 sβ1, find the rate of increase of the total surface area of the cube, in cm2 sβ1,
when π₯ = 8.
ππ
ππ‘= ..........................
Question 38 Categorisation: As above.
[Edexcel C4 June 2014 Q4]
A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When
the depth of the water is β cm, the volume of water π cm3 is given by
π = 4πβ(β + 4) , 0 β€ β β€ 25
Water flows into the vase at a constant rate of 80 π ππ 3π β1 .
Find the rate of change of the depth of the water, in ππ π β1 , when β = 6 .
πβ
ππ‘= .......................... πππ β1
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Answers
Question 1
Question 2
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Question 5
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Question 12
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Question 26
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Question 28
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