Chain Rule

Differentiation of composite functions

Tier 1

Formulad/dx[f(g(x))] = f'(g(x)) · g'(x)

56 questions

Progress

0/56

completed

2024(3 questions)

2024/ONE(a)Achievement

Differentiate

f (x) = (4 - 9 x^{4})

.

*You do not need to simplify your answer.*

2024/ONE(c)Merit

For the function below, find the range of values of

x

for which the function is decreasing.

y = 3 (2 x - 7)^{2} + 60 ln x + 12, x > 0

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2024/TWO(b)Achievement

An object is travelling in a straight line. Its displacement, in metres, is given by the formula

s (t) = ln (3 t^{2} + 5 t + 2)

, where

t > 0

and

t

is time, in seconds.

Find the velocity of this object when

t = 1

second.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2023(5 questions)

2023/ONE(a)Achievement

Differentiate

y = 3 x - 2

.

*You do not need to simplify your answer.*

2023/ONE(c)Merit

The graph shows the curve

y = \frac{2}{( x + 1 ) ^{3}}

, along with the tangent to the curve drawn at

x = 1

Loading diagram...

A second tangent to this curve is drawn which is parallel to the first tangent shown in the diagram.

Find the

x

-coordinate of the point where this second tangent touches the curve.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2023/THREE(a)Achievement

Differentiate

y = ln (x^{2} - x^{4} + 1)

.

*You do not need to simplify your answer.*

2023/THREE(e)Excellence

A power line hangs between two poles.

The equation of the curve

y = f (x)

that models the shape of the
power line can be found by solving the differential equation:

a \frac{d ^{2} y}{d x ^{2}} = 1 + (\frac{d y}{d x})^{2}

Use differentiation to verify that the function

y = \frac{a}{2} (e^{\frac{x}{a}} + e^{- \frac{x}{a}})

satisfies the above differential equation, where

a

is a positive
constant.

2023/TWO(b)Achievement

Find the gradient of the tangent to the curve

y = cot (2 x)

at the point where

x = \frac{π}{12}

.

You must use calculus and show any derivatives that you need to find when solving this problem.

2022(6 questions)

2022/ONE(a)Achievement

Differentiate

y = (ln x)^{2}

.

*You do not need to simplify your answer.*

2022/ONE(c)Merit

The graph below shows the function

y = x + 2

, and the normal to the function at the point where
the function intersects the

y

-axis.

Loading diagram...

Find the coordinates of point P, the

x

-intercept of the normal.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2022/THREE(a)Achievement

Differentiate

y = e^{4 x}

.

You do not need to simplify your answer.

2022/THREE(d)Excellence

Find the

x

-value(s) of any stationary point(s) on the graph of the function

y = 9 x - 2 + \frac{3}{3 x - 1}

and
determine their nature.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2022/TWO(b)Achievement

Find the gradient of the tangent to the curve

y = (3 x^{2} - 2)^{3}

when

x = 2

.

You must use calculus and show any derivatives that you need to find when solving this problem.

2022/TWO(e)Excellence

The curve with the equation

(y - 5)^{2} = 16 (x - 2)

has a tangent of gradient 1 at point P.

Loading diagram...

This tangent intersects the

x

and

y

axes at points R and S respectively.

Prove that the length RS is

72

.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2021(2 questions)

2021/TWO(a)Achievement

Differentiate

f (x) = (1 - x^{2})^{5}

.

*You do not need to simplify your answer.*

2021/TWO(e)Excellence

The graph below shows the curve

y = 2 x - 4

, and the tangent to the curve at point P.
The tangent passes through the point

(- 2, 1)

Loading diagram...

Find the coordinates of point P.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2020(5 questions)

2020/ONE(a)Achievement

Differentiate

y = (3 x - x^{2})^{5}

.

*You do not need to simplify your answer.*

2020/ONE(b)Achievement

Find the gradient of the tangent to the curve

y = 3 sin 2 x + cos 2 x

at the point where

x = \frac{π}{4}

.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2020/THREE(a)Achievement

Differentiate

y = 3 ln (x^{2} - 1)

.

*You do not need to simplify your answer.*

2020/THREE(c)Merit

The normal to the graph of the function

y = 2 x + 1

at the point

(4, 3)

intersects the

x

-axis at point P.

Loading diagram...

Find the

x

-coordinate of point P.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2020/TWO(b)Achievement

The value of a car is modelled by the formula

V = 17000 e^{- 0.25 t} + 2000 e^{- 0.5 t} + 500

for

0 \leq t \leq 20

where

V

is the value of the car in dollars (\

), an d

t$ is the age of the car in years.

Calculate the rate at which the value of the car is changing when it is 8 years old.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2019(6 questions)

2019/ONE(a)Achievement

Differentiate

y = 3 x^{2} - 1

.

*You do not need to simplify your answer.*

2019/ONE(b)Achievement

Find the rate of change of the function

f (t) = 5 ln (3 t - 1)

when

t = 4

.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2019/THREE(a)Achievement

Differentiate

y = \frac{4}{sin x}

.

*You do not need to simplify your answer.*

2019/THREE(e)Excellence

The graph below shows the function

y = 2 36 - x^{2}

, and the tangent to that function at point P.
The tangent intersects the

x

-axis at the point

(8, 0)

Loading diagram...

Find the

x

-coordinate of point P.
*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2019/TWO(a)Achievement

Differentiate

y = (2 x - 5)^{4}

.

*You do not need to simplify your answer.*

2019/TWO(b)Achievement

Find the gradient of the tangent to the curve

y = tan 2 x

at the point on the curve where

x = \frac{π}{6}

.

You must use calculus and show any derivatives that you need to find when solving this
problem.

2018(6 questions)

2018/ONE(a)Achievement

Differentiate

y = 2 x^{3} + \frac{5}{( x ^{3} + 2 ) ^{3}}

*You do not need to simplify your answer.*

2018/ONE(b)Achievement

f (x) = 3 cos 3 x

, show that

9 f (x) + f^{''} (x) = 0

2018/ONE(c)Merit

Find the gradient of the curve

y = ln ∣ sin^{2} x ∣

at the point where

x = \frac{π}{6}

You must use calculus and show any derivatives that you need to find when solving this
problem.

2018/THREE(e)Excellence

Loading diagram...

The above shape is made from wire. It has both vertical and horizontal lines of symmetry.

The ends of the shape are at the vertices of a square with a side length of 10 cm, as shown in
the diagram above.

The length of the piece of wire through the centre of the shape is

x

cm.

Find the value(s) of

x

that enables the shape to be made with the minimum length of wire.
You do not need to prove that the length is a minimum.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2018/TWO(a)Achievement

Differentiate

y = 3 x + cosec 5 x .

2018/TWO(b)Achievement

A particle is travelling in a straight line. The distance, in metres, travelled by the particle may
be modelled by the function

s (t) = ln (3 t^{2} + 3 t + 1)

t \geq 0

where

t

is time measured in seconds.

Find the velocity of this particle after 2 seconds.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2017(3 questions)

2017/ONE(a)Achievement

Differentiate

y = x + tan (2 x)

2017/ONE(c)Merit

The normal to the parabola

y = 0.5 (x - 3)^{2} + 2

at the point

(1, 4)

intersects the parabola again
at the point P.

Loading diagram...

Find the

x

-coordinate of point P.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2017/TWO(a)Achievement

Differentiate

y = 2 (x^{2} - 4 x)^{5}

.

*You do not need to simplify your answer.*

2016(5 questions)

2016/ONE(b)Achievement

The height of the tide at a particular beach today is given by the function

h (t) = 0.8 sin (\frac{4 π}{25} t + \frac{π}{2})

where

h

is the height of water, in metres, relative to the mean sea level
and

t

is the time in hours after midnight.

Loading diagram...

At what rate was the height of the tide changing at that beach at 9.00 a.m. today?

2016/ONE(d)Merit

The tangents to the curve

y = \frac{1}{4} (x - 2)^{2}

at points P and Q are perpendicular.

Loading diagram...

Q is the point

(6, 4)

.

What is the

x

-coordinate of point P?
*You must use calculus and show any derivatives that you need to find when solving this problem.*

2016/THREE(a)Achievement

Differentiate

f (x) = 4 3 x + 2

2016/THREE(b)Achievement

Find the

x

-value at which a tangent to the curve

y = 6 x - e^{3 x}

is parallel to the

x

-axis.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2016/TWO(b)Achievement

Find the gradient of the tangent to the function

y = 2 x - 1

at the point

(5, 3)

.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2015(5 questions)

2015/ONE(a)Achievement

Differentiate

y = 6 tan (5 x)

2015/ONE(b)Achievement

Find the gradient of the tangent to the function

y = (4 x - 3 x^{2})^{3}

at the point

(1, 1)

.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2015/ONE(c)Merit

Find the values of

x

for which the function

f (x) = 8 x - 3 + \frac{2}{x + 1}

is increasing.

You must use calculus and show any derivatives that you need to find when solving this
problem.

2015/THREE(a)Achievement

For what value(s) of

x

does the tangent to the graph of the function

f (x) = 5 ln (2 x - 3)

have a
gradient of

4

?

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2015/TWO(a)Achievement

Differentiate

f (x) = 5 x - 3 x^{2}

2014(5 questions)

2014/ONE(a)Achievement

Differentiate

y = 5 cos (3 x)

2014/ONE(b)Achievement

Find the gradient of the normal to the function

y = (3 x^{2} - 5 x)^{2}

at the point

(1, 4)

.

*Show any derivatives that you need to find when solving this problem.*

2014/ONE(d)Merit

Find the

x

-value at which the tangent to the function

y = \frac{4}{e ^{2 x - 2}} + 8 x

is parallel to the

x

-axis.

*Show any derivatives that you need to find when solving this problem.*

2014/THREE(a)Achievement

Differentiate

y = (3 x^{2} + 4 x)^{2}

2014/TWO(b)Achievement

Find the gradient of the curve defined by

y = 8 ln (3 x - 2)

at the point where

x = 2

.

Show any derivatives that you need to find when solving this problem.

2013(5 questions)

2013/ONE(a)Achievement

Differentiate

y = tan (x^{2} + 1)

.

*You do not need to simplify your answer.*

2013/ONE(b)Achievement

Find the gradient of the tangent to the function

f (x) = ln (3 x - e^{x})

at the point where

x = 0

2013/THREE(b)Achievement

For the function

f (x) = x + \frac{16}{x - 2}

, find the

x

-values of any stationary points.

*You must use calculus and clearly show your working, including any derivatives you need to find when solving this problem.*

2013/TWO(a)Achievement

\nDifferentiate

y = 3 π - x^{2}

.\n\n*You do not need to simplify your answer.*

2013/TWO(b)Achievement

A curve has the equation

y = (x^{3} - 2 x)^{3}

.

Find the equation of the tangent to the curve at the point where

x = 1

.

*Show any derivatives that you need to find when solving this problem.*

Back to Path Next: Graph Analysis