Independent preparation for the SQA Advanced Higher Mathematics exam. Not affiliated with or endorsed by Qualifications Scotland (SQA).

Section 2: Differentiation

Name the rule before you use it: chain, product or quotient, and for implicit differentiation, multiply each y term's derivative by dy/dx.

A lesson for this section

Quick checks: multiple choice

Question 1Paper 1 style · no calculator

Given

y

= \frac{e^{3x}}{x^2 + 1}

, find

\frac{dy}{dx}

Question 2Paper 1 style · no calculator

Differentiate

\sec 2x

with respect to

x

Question 3Paper 1 style · no calculator

Given

y

= e^{\tan(x^2)}

, find

\frac{dy}{dx}

Question 4Paper 2 style · calculator

The curve

y

= e^{2x}\cos x

has exactly one stationary point with

0

< x

< \frac{\pi}{2}

. Find its

x

coordinate, in radians to 3 significant figures.

Question 5Paper 1 style · no calculator

Given

y

= \cot 4x

, find

\frac{dy}{dx}

Question 6Paper 1 style · no calculator

A curve is given by

x

= y^3 + 4y

. Find the value of

\frac{dy}{dx}

at the point where

y

= 1

Question 7Paper 1 style · no calculator

Given

y

= \sin^{-1}(3x)

, find

\frac{dy}{dx}

Question 8Paper 1 style · no calculator

Given

y

= \tan^{-1}\sqrt{x}

x

> 0

, find

\frac{dy}{dx}

Question 9Paper 2 style · calculator

Find the gradient of the curve

y = \cos^{-1}\left(\frac{x}{2}\right)

at the point where

x

= 1

, to 3 significant figures.

Question 10Paper 1 style · no calculator

Find the equation of the tangent to the curve

y

= x^2\tan^{-1} x

at the point where

x

= 1

Question 11Paper 1 style · no calculator

The curve

x^2 + 3xy - y^2

= 9

passes through the point

(2, 1)

. Find

\frac{dy}{dx}

at this point.

Question 12Paper 1 style · no calculator

For the curve

x^2 + 4y^2

= 20

, find the value of

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

at the point

(2, 2)

Question 13Paper 1 style · no calculator

Given

y

= x^{2x}

x

> 0

, find

\frac{dy}{dx}

Question 14Paper 2 style · calculator

Given

y

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

x

> 2

, find the value of

\frac{dy}{dx}

when

x

= 4

, to 3 significant figures.

Question 15Paper 1 style · no calculator

A curve is defined by

x

= t^2 + 1

y

= t^3 - 2t

. Find

\frac{dy}{dx}

when

t

= 2

Question 16Paper 1 style · no calculator

A curve is defined by

x

= t^2

y

= t^3 + t

t

> 0

. Find the value of

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

when

t

= 1

Question 17Paper 2 style · calculator

A particle moves so that its position at time

t

seconds is

x

= 3\cos 2t

y

= 4\sin t

, with distances in metres and angles in radians. Find its speed when

t

= 1

, to 3 significant figures.

Question 18Paper 1 style · no calculator

A curve is defined by

x

= t + \frac{1}{t}

y

= t - \frac{1}{t}

t

> 0

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

t

= 2

Question 19Paper 2 style · calculator

A spherical balloon is inflated so that its volume increases at

50

^3

per second. Find the rate at which its radius is increasing when the radius is

6

cm, to 3 significant figures. (The volume of a sphere is

V

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

Question 20Paper 1 style · no calculator

A cylinder of height

h

cm fits exactly inside a sphere of radius

3

cm, so that the circles at both ends of the cylinder lie on the sphere. Find the greatest possible volume of the cylinder.

Question 21Paper 2 style · calculator

The concentration of a drug in the blood,

t

hours after a dose, is

C(t)

= 20te^{-\frac{t}{4}}

units. Find the greatest concentration, to 3 significant figures. You may need

e^{-1}

\approx 0.3679

Question 22Paper 1 style · no calculator

The point

(2, 4)

lies on the curve

x^3 + y^3

= 9xy

. Find the equation of the tangent to the curve at this point.

Question 23Paper 2 style · calculator

A ladder

6.5

m long leans against a vertical wall on level ground. Its foot slides away from the wall at

0.3

m/s. How fast is the top of the ladder sliding down the wall when the foot is

2.5

m from the wall?

Question 24Paper 2 style · calculator

A particle moves so that its position at time

t

seconds is

x

= 3t^2 - t^3

y

= 4t

, with distances in metres. Find its greatest speed for

0

\leq t

\leq 3

, to 3 significant figures, and the time at which it occurs.

Question 25Paper 1 style · no calculator

Sand falls onto a pile in the shape of a cone whose base radius is always twice its height. The volume of sand increases at

12\pi

^3

per second. Find the rate at which the height is increasing when the height is

2

cm. (The volume of a cone is

V

= \frac{1}{3}\pi r^2h

Question 26Paper 2 style · calculator

Find the least value of

y

= x^{x}

for

x

> 0

, to 3 significant figures.

Question 27Paper 1 style · no calculator

Given

y

= x\tan^{-1} x

{}- \frac{1}{2}\ln(1 + x^2)

, find the value of

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

when

x

= \sqrt{3}

Question 28Paper 2 style · calculator

Find the shortest distance from the point

(3, 0)

to the curve

y

= \sqrt{x}

, to 3 significant figures.

Paper-style questions: Paper 1, then Paper 2

An implicit curvePaper 1 style, no calculator · 12 marks

A curve has equation

x^2 - xy + y^2

= 12

(a) Find the value of

\frac{dy}{dx}

at the point

(-2, 2)

. [3]

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

(b) Find the equation of the tangent to the curve at

(-2, 2)

. [2]

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

x

coordinate. Give your answer as

(x, y)

. [3]

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

(d) Find the value of

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

at the point

(2, 4)

. [3]

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

(e) State what your answer to the previous part tells you about the point

(2, 4)

on the curve. [1]

0 words · a sentence

A particle in a planePaper 2 style, calculator · 11 marks

A particle moves in a plane so that its position at time

t

seconds is given by

x

= 1 + 4t - t^2

y

= 2\ln(1 + t)

t

\geq 0

, with distances in metres.

(a) Find the speed of the particle when

t

= 1

. Give your answer to 3 significant figures. [3]

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

(b) Find the value of

t

when the particle is moving parallel to the

y

-axis. [2]

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

y

coordinate of the particle at that time. Give your answer to 3 significant figures. [1]

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

(d) Find

\frac{dy}{dx}

in terms of

t

. [2]

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

(e) Find the value of

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

when

t

= 1

. [3]

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