Section 7: Differentiation

Rewrite roots and fractions as powers of x before you differentiate, and use the chain rule on every bracket.

Quick checks: multiple choice

Question 1Paper 1 style · no calculator

Given

f(x)

= 3x^4 - \frac{2}{x}

x

\neq 0

, find

f'(x)

Question 2Paper 1 style · no calculator

A curve has equation

y

= \sqrt{x}\,(x - 3)

x

> 0

. Find the gradient of the curve at the point where

x

= 4

Question 3Paper 1 style · no calculator

Given

f(x)

= \frac{2x^3 + 5}{x}

x

\neq 0

, find

f'(x)

Question 4Paper 2 style · calculator

Find the

x

coordinates of the points on the curve

y

= x^3 - 2x

where the gradient is

5

. Give your answers to 2 decimal places.

Question 5Paper 1 style · no calculator

Differentiate

3\sin x - 2\cos x

with respect to

x

Question 6Paper 1 style · no calculator

Given

y

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

, find

\frac{dy}{dx}

Question 7Paper 1 style · no calculator

Given

f(x) = \cos\left(4x - \frac{\pi}{3}\right)

, find the exact value of

f'\left(\frac{\pi}{4}\right)

Question 8Paper 1 style · no calculator

Given

y

= \sin^3 x

, find

\frac{dy}{dx}

Question 9Paper 2 style · calculator

Given

f(x)

= \sqrt{x^2 + 3}

, find

f'(1.5)

, giving your answer to 3 significant figures.

Question 10Paper 1 style · no calculator

Find the equation of the tangent to the curve

y

= x^3 - 2x^2 + 1

at the point where

x

= 2

Question 11Paper 1 style · no calculator

Find the equation of the tangent to

y

= 2\sin x

at the point where

x

= \frac{\pi}{6}

(

x

in radians).

Question 12Paper 1 style · no calculator

The tangent to the curve

y

= x^2 - 6x + 10

at the point P is parallel to the line

y

= 2x + 5

. Find the coordinates of P.

Question 13Paper 2 style · calculator

Find the equation of the tangent to

y

= 3\cos 2x

at the point where

x

= 0.5

(

x

in radians). Give the gradient and the intercept to 3 significant figures.

Question 14Paper 1 style · no calculator

For which values of

x

f(x)

= x^3 - 3x^2 - 9x + 4

strictly decreasing?

Question 15Paper 1 style · no calculator

Find the stationary points of

y

= 2x^3 + 3x^2 - 12x

and determine their nature.

Question 16Paper 1 style · no calculator

The curve

y

= x^3 - 6x^2 + 12x - 5

has exactly one stationary point. Find its coordinates and nature.

Question 17Paper 1 style · no calculator

Which describes the behaviour of

y

= 5x^2 - 2x^3 + x

for large positive and large negative values of

x

Question 18Paper 2 style · calculator

Find the

x

coordinate of the minimum turning point of

y

= x^3 - 3x^2 - 6x + 2

, to 2 decimal places.

Question 19Paper 1 style · no calculator

Find the greatest and least values of

f(x)

= 2x^3 - 9x^2 + 12x

on the interval

0

\leq x

\leq 3

Question 20Paper 2 style · calculator

f(x)

= x + 2\cos x

, where

x

is in radians. Find the least value of

f(x)

for

0

\leq x

\leq \pi

, giving your answer to 3 significant figures.

Question 21Paper 1 style · no calculator

A hill path is modelled by

h(x)

= -2x^3

{}+ 15x^2

{}- 24x

{}+ 30

for

0

\leq x

\leq 5

, where

h

is the height in metres above a lake and

x

is the horizontal distance in hundreds of metres from the start. Find the greatest height of the path.

Question 22Paper 2 style · calculator

A ball is thrown upwards. Its height,

s

metres, after

t

seconds is

s

= 20t - 4.9t^2

. Find the rate at which the height is changing when

t

= 1.5

Question 23Paper 2 style · calculator

The depth of water at a harbour wall,

D

metres,

t

hours after midnight is

D(t)

= 6 + 5\sin(0.4t)

, where

0.4t

is in radians. Find the rate of change of the depth when

t

= 3

, in metres per hour, to 3 significant figures.

Question 24Paper 1 style · no calculator

The volume of a sphere of radius

r

cm is

V

= \frac{4}{3}\pi r^3

. Find the rate of change of

V

with respect to

r

when

r

= 6

Question 25Paper 2 style · calculator

A closed cylindrical can holds

500

^3

. Its surface area, in cm

^2

, is

A(r)

= 2\pi r^2

{}+ \frac{1000}{r}

, where

r

cm is the radius. Find the radius that makes the surface area a minimum, to 3 significant figures.

Paper-style questions: Paper 1, then Paper 2

Sketching a quarticPaper 1 style, no calculator · 12 marks

A curve has equation

y

= 4x^3 - x^4

(a) Find the

x

coordinates of the points where the curve meets the

x

axis. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Find the

x

coordinates of the stationary points of the curve. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) The gradient of the curve is zero at the origin. Explain why the origin is not a turning point. [1]

0 words · a sentence

(e) Which describes the behaviour of $y$ for large positive and large negative values of $x$ ? [1]

(f) Find the equation of the tangent to the curve at the point where

x

= 1

. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

Making an open tankPaper 2 style, calculator · 10 marks

A metal tank is a cuboid with a square base of side

x

metres and height

h

metres, as shown. It has no lid. The tank must hold

5

^3

of water. The area of metal used is the base plus the four sides.

(a) Find an expression for

h

in terms of

x

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Hence find an expression for the area of metal,

A

square metres, in terms of

x

only. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

x

that makes the area of metal a minimum. Give your answer to 3 significant figures. [4]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Which of these justifies that this value of $x$ gives a minimum? [2]

(e) Find the least area of metal needed. Give your answer to 3 significant figures. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).