ONLINE: 4 UNIT MATHS 1

MATHEMATICS EXTENSION 2

4 UNIT MATHEMATICS

DIFFERENTIATION

Essential Background Topic

Differentiation is concerned with the rates of change of physical quantities. It is a

fundamental topic in mathematics, physics, chemistry, engineering etc.

Consider a continuous and single value function ( )y f x . The rate of change of y with

respect to x at the point x1 is called the derivative and equals the slope of the tangent

to the curve ( )y f x at the point x1. The process of finding the derivative of a function

is called differentiation.

Take two points P(x1, y1) and Q(x2, y2) on the curve ( )y f x . We require the slope of

the tangent at the point P(x1, y1). The slope of the straight line (chord) joining the

points P and Q is

2 1 2 1

2 1 2 1

( ) ( )slope PQ

y y f x f x

x x x x

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Let 2 1x x x ,

2 1x x x and 2 1( ) ( )f x f x x . The point Q approaches the point P as 0x and the

slope of the chord approaches the slope of the tangent at the point x1. Intuitively, we can say that the slope

of the tangent at P will be given by the limit of the slope of the chord as 0x

1 1

0

( ) ( )slope at P limit

x

f x x f x

x

Given a curve ( )y f x and a point P on the curve, the slope of the curve at P is the limit of the slope of lines

between P and Q on the curve as Q approaches P. The slope of a curve ( )y f x is the rate at which y is

changing as x changes or it is the rate of change of y with respect to x. This slope is known as the derivative

of the function y with respect to x. It is given by the special symbols

( )

'( )dy d f x

f x ydx dx

0

( ) ( )limit

x

dy f x x f x

dx x

ONLINE: 4 UNIT MATHS 3

RULES FOR DIFFERENTIATION

The derivative of a constant is zero

constanty

0dy

dx

ONLINE: 4 UNIT MATHS 4

The derivative of powers of x

1

n

n

y A x

dyn A x

dx

Example

3 2

2

2 5 8 12

/ 6 10 8

y x x x

dy dx x x

ONLINE: 4 UNIT MATHS 5

Proof

3 2

3 2

3 2 2 2

2

2

0

( ) 2 5 8 12

( ) 2 5 8 12

( ) 2 5 8 12 6 10 8 6 2 5

( ) ( )6 10 8 6 2 5

( ) ( )limit 6 10 8

x

f x x x x

f x x x x x x x x x

f x x x x x x x x x x x

f x x f xx x x x x

x

f x x f x dyx x

x dx

The derivative of a product

1 2

1 2 2 1

( ) ( )

( ) ( ) ( ) ( )

y f x f x

dy d df x f x f x f x

dx dx dx

1 2( ) ( )u f x v f x

y u v

dy dv duu v

dx dx dx

This rule can be extended to the product of several functions

1 2 3( ) ( ) ( )u f x v f x w f x

y u v w

dy dw dv duu v u w v w

dx dx dx dx

ONLINE: 4 UNIT MATHS 6

Example

6 1/ 2 2 1/ 2

6 1/ 2 5 3/ 2

2 1/ 2 1/ 2

6 1/ 2 1/ 2

2 1/ 2 5 3/ 2

7 1/ 2 11/ 2 1

7 11/ 2 5 1/ 2

3 4 3 6 8

3 4 / 18 2

3 6 8 / 6 3

/ 3 4 6 3

3 6 8 18 2

/ 18 24 9 12

54 108 144 6

y x x x x

u x x du dx x x

v x x dv dx x x

dy dv duu v

dx dx dx

dy dx x x x x

x x x x

dy dx x x x x

x x x x

1 3/ 2

7 11/ 2 5 1/ 2 3/ 2

12 16

/ 72 117 144 18 16

x x

dy dx x x x x x

ONLINE: 4 UNIT MATHS 7

The chain rule – differentiation of a function of a function

If ( )y f u and ( )u g x then the derivative of y with respect to x is given by

dy dy du

dx du dx

Example

7 / 22

2

7 / 2 5/ 2

5/ 2

5/ 22

2 3 5

2 3 5 / 4 3

7/2

74 3

2

72 3 5 4 3

2

y x x

u x x du dx x

y u dy du u

dy dy duu x

dx du dx

dyx x x

dx

ONLINE: 4 UNIT MATHS 8

The derivative of a quotient

The derivative of the quotient ( )

( ) ( )( )

f x uy u f x x g x

g x w provided that ( ) 0g x is given by

2

/ /w du dx u dw dxdy

dx w

I think it often better to use only the product rule and not the quotient rule

1

uy

w

wv

y u v

dy dv duu v

dx dx dx

Proof

1

2

2

1 1

/ /

dy d u du w

dx dx w dx

dy dw duu

dx w dx w dx

w du dx u dw dxdy

dx w

ONLINE: 4 UNIT MATHS 9

Example

1/ 21/ 2

1/ 2 1/ 2

1/ 2 3/ 2

1/ 2 3/ 2 1/ 2 1/ 2

3/ 2 1/ 2

2 1/ 2

2 1

2 1

2 12 1 2 1

2 1

2 1 / 2 1

2 1 / 2 1

2 1 2 1 2 1 2 1

2 1 2 1 2 1

2 1 2 1

xy

x

xy x x

x

u x du dx x

v x dv dx x

y u v

dy dv duu v

dx dx dx

dyx x x x

dx

x x xdy

dx x x

ONLINE: 4 UNIT MATHS 10

Differentiation of trigonometric functions

sin cosd

x xdx

cos sind

x xdx

2

2

1tan sec

cos

dx x

dx x

1

cosec cosec cotsin

d dx x x

dx dx x

1

sec sec tancos

d dx x x

dx dx x

21cot cosec

tan

d dx x

dx dx x

Examples

2

2

sin

sin / 2 / cos

/ / / 2 cos

/ 2sin cos

y x

u x y u dy du u du dx x

dy dx dy du du dx u x

dy dx x x

ONLINE: 4 UNIT MATHS 11

1 2

2 1

2

sin

sin / cos /

/ / /

/ sin cos

cos sin/

xy

x

u x du dx x v x dv dx x

y u v dy dx u dv dx v du dx

dy dx x x x x

x x xdy dx

x

Differentiation of inverse trigonometric functions

If ( )y f x then the inverse function is ( )x g y then

( ) 1 1

( ) / /

dg y dx

dy df x dx dy dy dx

-1

2 2

1sin

d x

dx a a x

-1

2 2

1cos

d x

dx a a x

-1

2 2tan

d x a

dx a a x

ONLINE: 4 UNIT MATHS 12

Examples

-1

2 2

2 2

sin

/ sin sin

/ cos

1/ 1/ /

cos

cos

1/

xy

a

x a y x a y

dx dy a y

dy dx dx dya y

a xy

a

dy dxa x

-1

2 2

2 2

cos

/ cos cos

/ sin

1/ 1/ /

sin

sin

1/

xy

a

x a y x a y

dx dy a y

dy dx dx dya y

a xy

a

dy dxa x

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-1

1 2

2 1

2 2

2

2

2

2 2

tan

sintan

cos

sin / cos cos / sin cos

/ / /

/ sin sin cos cos cos

sin cos/

cos

/cos

cos/

/

xy

a

a yx a y

y

u a y du dx a y v y dv dx y y

x u v dx dy u dv dy v du dy

dx dy a y y y y a y

y ydx dy a

y

adx dy

y

ydy dx

a

ady dx

a x

ONLINE: 4 UNIT MATHS 14

Differentiation of inverse exponential and logarithmic functions

/

b x

b x

y a e

dy dx a b e b y

log ln

/

ey a b x a b x

ady dx

x

/ log

x

x

e

y a

dy dx a a

Examples and proofs

/

/

/

/

log

1/

/ 1/ 1/

/

/

e

y a

y a

y a

y a

y a b x

e b x

x b e

dx dy b a e

a b a bdy dx

e b x

ady dx

x

ONLINE: 4 UNIT MATHS 15

log log log

log

log

1/

log

/ log

/ log

x

x

e e e

e

e

e

e

x

e

y a

y a x a

yx

a

dx dyy a

dy dx y a

dy dx a a

2

2

2

log 3 2

3 2 / 2 3

log / 1/

/ / /

/ 1/ 2 3

2 3/

3 2

e

e

y x x

u x x du dx x

y u dy du u

dy dx dy du du dx

dy dx u x

xdy dx

x x

ONLINE: 4 UNIT MATHS 16

log2 3

log log 2 3

1 3/

2 3

2/

2 3

e

e e

xy

x

y x x

dy dxx x

dy dxx x

Higher derivatives

We can differentiate a function many times

Function ( )y f x

1st derivative / '( )dy dx f x y

2nd derivative 2 2/ ''( )d y dx f x y

Example

5 4 3 2

4 3 2

2 2 3 2

3 3 2

4 4

5 5

6 6

2 3 4 5 6 9

/ 10 12 12 10 6

/ 40 36 24 10

/ 120 72 24

/ 240 72

/ 240

/ 0

y x x x x x

dy dx x x x x

d y dx x x x

d y dx x x

d y dx x

d y dx

d y dx

ONLINE: 4 UNIT MATHS 17

APPLICATIONS OF DIFFERENTIATION

The derivative of the function ( )y f x is the rate of change of y with respect to x. The derivative is a very

useful quantity in analysing systems that change with time.

In radioactive decay, the number of nuclei N remaining time t is

0

tN N e

where N0 is the number of nuclei at time t = 0 and is a constant called the decay constant.

The rate of decay is proportional to the number of remaining nuclei

0

tdNN e N

dt

The minus sign indicates the number of nuclei remaining decreases with time.

The displacement s of a moving particle is a function of time t.

displacement ( )s f t

velocity /v ds dt

acceleration 2 2/ /a d s dt dv dt

ONLINE: 4 UNIT MATHS 18

Example

A particle’s displacement s is a function of time t is given by the equation

3 24 3 6 1s t t t

At what time is the acceleration of the particle zero?

2/ 12 6 6

/ 24 6

6 10 24 6 0

24 4

v ds dt t t

a dv dt t

a t t