Quotient Rule

Differentiation of quotients

Tier 2

Formulad/dx[f/g] = (f'g - fg')/g²

17 questions

Progress

0/17

completed

2024(3 questions)

2024/THREE(c)Merit

Find the

x

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

f (x) = \frac{x ^{2} - 5 x + 4}{x ^{2} + 5 x + 4}

.

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

You do not need to determine the nature of any stationary point(s) found.

2024/TWO(d)Merit

Consider the function

f (x) = \frac{ln x}{x}

x > 0

.

Find the coordinates of the point of inflection on the graph of the function.

*You can assume that your point found is actually a point of inflection.*

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

2024/TWO(e)Excellence

The graph of the function

y = \frac{x e ^{3 x}}{2 x + k}

, where

k

is a non-zero constant, has a single turning point at Q.

Find the

x

-coordinate of the point Q.

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

2023(2 questions)

2023/TWO(a)Achievement

Differentiate

f (x) = \frac{x ^{2}}{cos x}

.

*You do not need to simplify your answer.*

2023/TWO(c)Merit

A curve is defined by the equation

f (x) = \frac{e ^{x}}{x ^{2} + 2 x}

.

Find the

x

-value(s) of any point(s) on the curve where the tangent to the curve is parallel to the

x

-axis.

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

2022(1 questions)

2022/TWO(c)Merit

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

d (t) = \frac{t ^{2} - 6}{2 t ^{3}}

where

t > 0

t

is time in seconds.

Find the time(s) when the object is stationary.

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

2021(4 questions)

2021/THREE(a)Achievement

Differentiate

y = \frac{cot x}{x ^{2} + 1}

.

You do not need to simplify your answer.

2021/THREE(c)Excellence

For what values of

x

is the function

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

increasing?

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

2021/THREE(d)Merit

A curve has the equation

y = \frac{4 x + k}{4 x - k}

, where

k

is a constant and

x \neq = \frac{k}{4}

.

The point P lies on the curve and has an

x

-coordinate of 3.

The gradient of the tangent to the curve at P is

\frac{- 8}{27}

.

Find the possible value(s) of

k

.

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

2021/TWO(b)Achievement

A curve has the equation

y = \frac{x ^{2}}{x + 1}

.

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.*

2020(2 questions)

2020/ONE(c)Merit

Find the value of

x

for which the graph of the function

y = \frac{x}{1 + ln x}

has a stationary point.

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

2020/TWO(a)Achievement

Differentiate

y = \frac{tan x}{x ^{3}}

.

*You do not need to simplify your answer.*

2017(1 questions)

2017/ONE(e)Excellence

Find the values of

a

and

b

such that the curve

y = \frac{a x - b}{x ^{2} - 1}

has a turning point at

(3, 1)

.

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

2016(2 questions)

2016/THREE(d)Merit

y = \frac{e ^{x}}{sin x}

, show that

\frac{d y}{d x} = y (1 - cot x)

2016/THREE(e)Excellence

In a rugby game, a try is scored 15 m from the left-hand goal-post. The conversion kick is
taken at some point on the line perpendicular to the goal-line from the point where the try was
scored, as shown in the diagram below.

The ball needs to pass between the goal-posts, which are 5.4 m apart.

Loading diagram...

Find the distance

d

from the goal-line that the conversion kick should be taken from in order
to maximise the angle

θ

between the lines from the ball to the goal-posts.

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

*You do not need to prove that the angle you have found is a maximum.*

2015(2 questions)

2015/ONE(d)Merit

For what value(s) of

x

is the tangent to the graph of the function

f (x) = \frac{x + 4}{x ( x - 5 )}

parallel to
the

x

-axis?

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

2015/THREE(b)Achievement

f (x) = \frac{x}{e ^{3 x}}

, find the value(s) of

x

such that

f^{'} (x) = 0

.

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

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