Chain + Product

Combined chain rule and product rule

Tier 3

21 questions

Progress

0/21

completed

2024(4 questions)

2024/ONE(d)Merit

Find the

x

-value(s) of any stationary points on the graph of the function below, and determine
their nature.

y = (2 x - 1) e^{- 2 x}

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

2024/THREE(a)Achievement

Differentiate

y = x . sec (6 x)

.

You do not need to simplify your answer.

2024/THREE(e)Excellence

The diagram below shows part of the graph of the function

f (x) = e^{- x^{2}}

, where

x \geq 0

Loading diagram...

The point P lies on the curve and the point Q lies on the

x

-axis so that OP = PQ, where O is
the origin.

Prove that the largest possible area of the triangle OPQ is

\frac{1}{2 e}

.

*You do not need to show that the area you have found is a maximum.*

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

2024/TWO(c)Merit

Show that

y = sin (x^{2}) - cos (x)

is a solution to the equation

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

2023(2 questions)

2023/ONE(b)Achievement

Find the rate of change of the function

f (t) = t^{2} e^{2 t}

when

t = 1.5

.

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

2023/ONE(e)Excellence

The graph of

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

, where

m > 0

, is shown.

Loading diagram...

The total shaded area between the curve and the

x

-axis
from

x = 0

x = 2 m

is given by

A = \frac{4 m ^{4}}{3}

.

A right-angled triangle is now constructed with one
vertex at

(0, 0)

and another on the curve

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

,
as shown below.

Loading diagram...

Show that the maximum area of such a triangle is

\frac{3}{8}

of the total shaded area.

*You must use calculus and show any derivatives that you need to find when solving this problem.*
*You do not have to prove that the area you have found is a maximum.*

2022(2 questions)

2022/ONE(e)Excellence

p

is a positive real constant, prove that

y = e^{p x^{2}}

does not have any points of inflection.

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

2022/TWO(a)Achievement

Differentiate

f (x) = (5 x - 3) sin (4 x)

.

*You do not need to simplify your answer.*

2021(4 questions)

2021/ONE(a)Achievement

Differentiate

y = e^{3 x} sin 2 x

.

*You do not need to simplify your answer.*

2021/ONE(c)Merit

A curve has the equation

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

.

Find the

x

-coordinate(s) of any stationary point(s) on the curve.

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

2021/ONE(d)Excellence

A curve is defined parametrically by the equations

x = t^{2} + 3 t

and

y = t^{2} ln (2 t - 3)

, for

t > \frac{3}{2}

.

Find the gradient of the tangent to the curve at the point

(10, 0)

.

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

2021/TWO(c)Merit

A curve has the equation

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

.

Find the equation of the normal to the curve at the point where the curve crosses the

y

-axis.

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

2020(2 questions)

2020/THREE(e)Excellence

A curve has the equation

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

.

Prove that

\frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} + 4 y = 0

.

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

2020/TWO(c)Merit

Find the

x

-coordinates of any stationary points on the graph of the function

f (x) = (2 x - 3) e^{x^{2} + k}

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

2019(1 questions)

2019/TWO(e)Excellence

y = e^{u}

and

u = sin 2 x

show that

\frac{d ^{2} y}{d x ^{2}} = \frac{d ^{2} y}{d u ^{2}} (\frac{d u}{d x})^{2} + \frac{d y}{d u} \cdot \frac{d ^{2} u}{d x ^{2}}

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

2017(3 questions)

2017/THREE(a)Achievement

Differentiate

y = x ln (3 x - 1)

.

*You do not need to simplify your answer.*

2017/THREE(e)Excellence

For the function

y = e^{x} cos k x

:

(i) Find

\frac{d y}{d x}

and

\frac{d ^{2} y}{d x ^{2}}

.

(ii) Find all the value(s) of

k

such that the function

y = e^{x} cos k x

satisfies the equation

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + 2 y = 0

for all values of

x

2017/TWO(e)Excellence

A rectangle is inscribed in a semi-circle of radius

r

, as shown below.

Loading diagram...

Show that the maximum possible area of such a rectangle occurs when

x = \frac{r}{2}

.

*You do not need to prove that your solution gives the maximum area.*

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