Ah unit 1 differentiation

Unit 1
Differentiation
1 Differentiation from first principles
Definition:
( ) ( )
( ) lim
0
f x h f x
f x
hh
3
2
3
3 2 2
3 2 2 3
Find, from first principles, the derivative of:
4
(a) 3 (b)
(a) ( ) 3
( ) 3( ) 3( )( 2 )
3( 3 3 )
x
x
f x x
f x h x h x h x hx h
x hx h x h
3 2 2 3
3 2 2 3 3
2 2 3 2 2
2 2
2 2
3 9 9 3
( ) ( ) 3 9 9 3 3
9 9 3 (9 9 3 )
( ) ( ) (9 9 3 )
9 9 3
x hx h x h
f x h f x x hx h x h x
hx h x h h x hx h
f x h f x h x hx h
x hx h
h h
2 2 2
0 0
( ) ( )
( ) lim lim 9 9 3 9 .
h h
f x h f x
f x x hx h x
h

2
2
2 2
2 2
2 2 2 2
2 2 2
2 2
4
(b) ( )
4
( )
( )
4 4
( ) ( )
( )
4 4( )
=
( ) ( )
4 4( 2 )
( )
f x
x
f x h
x h
f x h f x
x h x
x x h
x x h x x h
x x hx h
x x h
2
2 2
2 2
2 2 2 2
2 20 0
8 4
( )
( 8 4 )
( )
( ) ( ) ( 8 4 ) 1 8 4
( ) ( )
( ) ( ) 8 4 8
( ) lim lim
( )h h
hx h
x x h
h x h
x x h
f x h f x h x h x h
h x x h h x x h
f x h f x x h x
f x
h x x h 2 2 4 3
8 8
.
x
x x x x
2 3
2
Exercise 1
Differentiate from first principles:
4 1
(a) 6 (b) 4 (c) (d)x x
x x

2 Reminders from Higher
1
Basic facts
( ) , ( )
( ) sin , ( ) cos
( ) cos , ( ) sin
n n
f x ax f x nax
f x x f x x
f x x f x x
The Chain Rule…….extremely important and often forgotten.
The chain rule must be applied whenever the question involves a composition of functions
or a “function of a function”.
2 2
2 1
2 2 2
Examples
10 10 1
(a) ( ) . ( )
7(3 8) 7 (3 8)
10
(3 8)
7
10
( ) (3 8) (3 8)
7
f x f x
x x
x
d
f x x x
dx
2 2
2 2
2
10
(3 8) 6
7
60
(3 8)
7
60 1
7 (3 8
x x
x
x
x
x 2
2 2
)
60
( ) .
7(3 8)
x
f x
x
3 3 3
(b) 4cos3 . 4cos3
4sin3 (3 )
12sin 3
(c) ( ) 2sin . ( ) 2sin 2(sin )
y x y x
dy d
x x
dx dx
x
f x x f x x x
2
2
2
( ) 6(sin ) . (sin )
6(sin ) cos
( ) 6sin cos
d
f x x x
dx
x x
f x x x

4 4
4 3
3 4
(d) 3cos . 3cos
3sin .4
12 sin .
y x y x
dy
x x
dx
x x
Knowledge of Higher trig identities is required.
2 2
2 2
2
sin
cos sin 1 tan
cos
cos( ) cos cos sin sin
cos( ) cos cos sin sin
sin( ) sin cos cos sin
sin( ) sin cos cos sin
sin 2 2sin cos
cos2 cos sin
2cos 1
x
x x x
x
x y x y x y
x y x y x y
x y x y x y
x y x y x y
x x x
x x x
x
2
1 2sin x
Consider this example.
2
2 1
( ) 4cos . Find ( ), the second derivative (ie the derivative
of the derivative).
( ) 4(cos ) . ( ) 8(cos ) . (cos )
8cos
f x x f x
d
f x x f x x x
dx
x.( sin )
8cos sin
[At this point, to find the second derivative would not be possible unless we use
our trig identities].
x
x x
8cos sin
4(2sin cos )
4sin 2
( ) 4cos2 . (2 )
x x
x x
x
d
f x x x
dx
8cos2 .x

Exercise 2
Differentiate the following:
3 4
6
2
22
4
(a) ( ) 2 7 (b) ( ) 3 1 (c)
(5 6)
2 3
(d) (e) ( ) cos 4 (f)
cos2 3
f x x f x x y
x
y f x x y
xx
Reciprocal Trig functions and their derivatives
Definitions
1
secant of , written as sec
cos
1
cosecant of , written as cosec
sin
1 cos
cotangent of , written as cot
tan sin
x x
x
x x
x
x
x x
x x


 
Derivative of sec x.
1
2
2
2
1
( ) sec (cos )
cos
1
( ) (cos ) . (cos ) . sin
cos
sin
cos
sin 1
cos cos
f x x x
x
d
f x x x x
dx x
x
x
x
x x
tan sec .x x
(sec ) tan sec
d
x x x
dx
Exercise 3
Demonstrate the proof of the following result:
(cosec ) cot cosec
d
x x x
dx

The Product Rule
0 0
0
( ) ( ) ( ), ( ) ( ) ( ) ( ) ( )
Proof
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) lim lim
( ) ( ) (
lim
h h
h
f x u x v x f x u x v x u x v x
f x u x v x
f x h u x h v x h
f x h f x u x h v x h u x v x
f x h f x u x h v x h u x v x
f x
h h
u x h v x h u
0
0 0 0 0
) ( ) ( ) ( ) ( ) ( )
( ( ) ( )) ( ) ( )( ( ) ( ))
lim
( ( ) ( )) ( ( ) ( ))
lim lim ( ) lim ( )lim
( ) ( ) ( ) ( ) ( ).
h
h h h h
x v x h u x v x h u x v x
h
u x h u x v x h u x v x h v x
h
u x h u x v x h v x
v x h u x
h h
f x u x v x u x v x
In practice, use “differentiate the first, leave the second plus leave the first, differentiate
the second”.
Examples
3
1
2 3 2
1 1
2 2 2
(a) sin (b) ( ) cos2
3 sin cos cos2
1
(3sin cos ) ( ) cos2 . sin 2 . (2 )
2
[Note th
y x x f x x x
dy
x x x x x x
dx
d
x x x x f x x x x x x
dx
1
2
1
2
1 1
e factorisation]. cos2 2 sin 2
2
cos2
2 sin 2 .
2
x x x
x
x
x x
x

The Quotient Rule
2
1
1 2
2
( ) ( ) ( ) ( ) ( )
( ) . ( ) .
( ) ( )
Proof (using the Product Rule).
( )
( ) ( ) ( )
( )
( ) ( ) ( ) ( ). ( ) ( )
1 1
( ). ( ). . ( )
( ) ( )
( )
( )
u x u x v x u x v x
f x f x
v x v x
u x
f x u x v x
v x
d
f x u x v x u x v x v x
dx
u x u x v x
v x v x
u x
v x 2
2 2
2
( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( )
( )
u x v x
v x
u x v x u x v x
v x v x
u x v x u x v x
f x
v x

-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-8
-6
-4
-2
2
4
6
8
10
12
14
16
18
x
y
x
y e
-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5
-8
-6
-4
-2
2
4
6
8
10
12
14
16
18
x
y
x
y e
The Exponential Function
1
0 ( )
1 or exp( ) is called the exponential function
2 2.71828...
1
Also, lim 1 2.71828.... .
1
3 1, ,
x
n
n
x x y x y
x
e x
e e
e
n
e e e e e
e
4 Graphs of and .x x
y e y e

5 The derivative of x
e .
Consider the function .x
y a
0 0 0
0
( )
( )
( ) ( )
( ) lim lim lim
( 1)
lim
x
x h
x h x x h x
h h h
x h
h
f x a
f x h a
f x h f x a a a a a
f x
h h h
a a
h
0 0
0
0
( 1)
lim lim
( 1)
lim
( 1)
Let us consider lim in more detail.
Taking 2 and 0.
h
x
h h
h
x
h
h
h
a
a
h
a
a
h
a
h
a h 00001, the above gives a value of 0.69314.....
Now taking 3 and 0.00001, the above gives a value of 1.09861.....
We can assume that there is a value of between 2 and 3 which will give
a value of th
a h
a
0
e above equal to 1.
Let 2.71828. Now the above gives a value of 1.000004327.
In fact if , then the above gives a value of 1.
( 1)
So if then lim
h
x x
h
a
a e
dy e
y e e
dx h
e 1 .x x
e
This gives the (remarkable) result that the derivative of , is .x x
e e

Examples
3 3
3
3
Find the derivativeof the following functions:
3
(a) 4 (b) ( ) (c) 4 cos2
3
(a) 4 (b) ( )
4 . (3 )
x x
x
x
x
x
y e f x y e x
e
y e f x
e
dy d
e x
dx dx
1
2
1
3 2
3
1
12 ( ) 3 .
2
3 1
2
x
x
x
x
e
d
e f x e x
dx
e
3
3 3
3 3
3
3
2
(c) 4 cos2
4 (3 )cos2 4 . sin 2 (2 )
12 cos2 8 sin 2
4 (3cos2 2sin 2 )
x
x
x x
x x
x
e
y e x
dy d d
e x x e x x
dx dx dx
e x e x
e x x
[Note the factorisation].
Derivative of the Natural Logarithm Function
The natural logarithm function is defined as ln logey x x.

If ln ie log then (Higher log work).
..........*
[Note that is the derivative of with respect to ].
1
So
y
e
y
y
y x y x x e
x e
dx
x y
dy
dx
e
dy
dy
dxdx
d
2
1 1
.........from *
This gives the extremely important result:
1
(ln ) .
Examples
Differentiate the following:
3ln(2 5)
(a) ( ) ln(3 ) (b)
(a)
y
e x
y
d
x
dx x
x
f x x y
x
2 2
2 2
2
2
( ) ln(3 ) An alternative method: ( ) ln(3 )
1
( ) . (3 ) ln3 ln (Higher Maths)
3
1
6
3
f x x f x x
d
f x x x
x dx
x
x
ln3 2ln (ln3 is a constant)
2 1
( ) 0 2
x
f x
x x
1
2
1 1
2 2
21
2
1
2
1
2
2
3ln(2 5) 3ln(2 5)
(b)
1 1
3 . (2 5). 3ln(2 5).
2 5 2
6 1
. 3ln(2 5).
2 5
2
x
x x
y
x
x
d
x x x x
dy x dx
dx
x
x x
x
x
x
6 3ln(2 5)
2 5 2 Note this must be simplified.
x x
x x
x

6 3ln(2 5)
12 3(2 5)ln(2 5) 12 5 2( )
2(2 5) 2(2 5)
12 3(2 5)ln(2 5) 1
2(2 5)
x x
x x x xx xf x
x xx x x x
x x x
xx x
3
2
12 3(2 5)ln(2 5)
2(2 5)
x x x
x x

Ah unit 1 differentiation

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Ah unit 1 differentiation