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 fx . The rate of change of y with respect to x at the point x 1 is called the derivative and equals the slope of the tangent to the curve () y fx at the point x 1 . The process of finding the derivative of a function is called differentiation. Take two points P(x 1 , y 1 ) and Q(x 2 , y 2 ) on the curve () y fx . We require the slope of the tangent at the point P(x 1 , y 1 ). 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 fx fx x x x x
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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
ONLINE: 4 UNIT MATHS 2
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
ONLINE: 4 UNIT MATHS 13
-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
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