2. Differentiation
Today’s Objective:
When you have completed this lecture you will be able to:
• Differentiate by using a list of standard derivatives
• Apply the chain rule
• Apply the product and quotient rules
• Perform logarithmic differentiation
• Differentiate parametric equations
2
3. Here is a revision list of the standard derivatives which you have no
doubt used many times before. Memorize those with which you
are less familiar.
3
Standard derivatives
No.
1
2
3
4
5
6
7
x
sin
No.
8
9
10
11
12
13
14
)
(x
f
y
dx
dy
n
x 1
n
nx
x
e x
e
x
a a
ax
ln
.
x
ln
x
1
x
a
log
a
x ln
.
1
x
sin x
cos
)
(x
f
y
dx
dy
x
2
sec
x
cot x
ec2
cos
x
sec x
x tan
.
sec
ecx
cos x
ecx cot
.
cos
x
sinh x
cosh
x
cosh x
sinh
x
tan
x
cos x
sin
kx
e kx
ke
4. 4
Example 1: write down the derivatives for the following.
Solution :
x
x
x
x
e
x
x 2
.
6
tan
.
5
ln
.
4
.
3
sin
.
2
.
1 3
5
x
e
x
x
x x
cos
.
11
.
10
log
.
9
cosh
.
8
sec
.
7 10
x
a
x
a
ecx
x .
16
cot
.
15
.
14
cos
.
13
sinh
.
12 3
2
4
.
20
.
19
log
.
18
.
17
x
a e
x
x
x
2
ln
.
2
.
6
sec
.
5
1
.
4
3
.
3
cos
.
2
5
.
1 2
3
4 x
x
x
x
e
x
x
x
e
x
x
x
x x
sin
.
11
.
10
10
ln
.
1
.
9
sinh
.
8
tan
.
sec
.
7
x
ec
anx
ecx
x 2
cos
.
15
0
.
14
cot
.
cos
.
13
cosh
.
12
2
2
1
5
.
20
2
1
.
19
ln
1
.
18
4
.
17
ln
.
.
16
x
x
e
x
a
x
x
a
a
5. 5
Function of a function
Is a function of since the value of depends on the value of
angle . Similarly is a function of the angle since
the value of the sine depends on the value of this angle.
i.e. Is a function of
But it itself is a function of , since its value depends on .
For example :
the derivative of the function
x
sin x x
sin
x )
5
2
sin(
x )
5
2
(
x
)
5
2
sin(
x
)
5
2
(
x
)
5
2
(
x
x x
5
)
3
4
(
x
y
4
)
3
4
(
5 x
dx
dy
)
3
4
(
x
4
)
3
4
(
5 4
x
dx
dy
4
)
3
4
(
20
x
6. 6
Example 2 : write down the derivatives for the following.
Solution :
)
cos
4
3
ln(
.
5
)
cos(
.
4
2
sin
.
3
.
2
)
5
4
(
.
1 2
3
6
x
y
x
y
x
y
e
y
x
y x
x
y
x
y
x
y
e
y
x
y x
3
cos
ln
.
10
)
3
(
cos
.
9
sin
.
8
.
7
)
1
2
(
log
.
6 3
2
2
sin
10
x
x
y
x
x
y
x
dx
dy
e
dx
dy
x
dx
dy x
cos
4
3
sin
4
.
5
)
sin(
2
.
4
2
cos
2
.
3
.
2
)
5
4
(
24
.
1
2
3
5
x
x
x
y
x
x
y
x
x
y
e
x
y
x
y x
3
tan
3
3
cos
3
sin
3
.
10
)
3
(
cos
)
3
sin(
9
.
9
sin
cos
2
.
8
.
2
cos
2
.
7
10
ln
)
1
2
(
2
.
6
2
2
sin
7. 7
Differentiate functions which are products or quotients of two of the functions
dx
du
v
dx
dv
u
dx
dy
1. Products : if y=uv where u and v are functions, of x, then
you already know that:
2
v
dx
dv
u
dx
du
v
dx
dy
2. Quotients : if y=u/v where u and v are functions, of x, then
you already know that:
8. 8
Example 3: write down the derivatives for the following.
Solution :
2
2
2
3
5
2
2
cos
.
8
ln
.
7
1
3
sin
.
6
sinh
ln
.
5
5
sin
.
4
2
cos
.
3
)
1
3
(
.
2
tan
.
1
x
x
y
e
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
e
y
x
x
y
x
x
)
1
3
(
5
3
.
2
tan
2
sec
.
1
5
5
2
2
x
e
e
dx
dy
x
x
x
x
dx
dy
x
x
)
5
15
(
3 5
5
x
e
e x
x
)
8
15
(
)
5
15
3
( 5
5
x
e
x
e x
x
x
x
x
dx
dy
2
cos
2
sin
2
.
3
x
x
x
x
dx
dy
5
sin
3
5
cos
5
.
4 2
3
)
5
sin
3
5
cos
5
(
2
x
x
x
x
x
x
x
x
x
dx
dy
sinh
2
sinh
cosh
.
5
2
)
sinh
2
coth
( x
x
x
x
2
)
1
(
3
sin
3
cos
)
1
(
3
.
6
x
x
x
x
dx
dy
x
x
x
x
e
x
x
e
x
e
x
e
dx
dy
2
4
2
2
ln
2
1
ln
2
.
7
4
2
2
cos
2
2
sin
2
.
8
x
x
x
x
x
dx
dy
3
)
2
cos
2
sin
(
2
x
x
x
x
9. 9
Logarithmic differentiation.
The rules of differentiating a product or a quotient that we are
revised are used when there are just two-factor functions, i.e. uv
or u/v. where there are more than two functions in any
arrangement top or bottom, the derivative is best found by what
is known as logarithmic differentiation. For example;
if where u, v, and w -and also y- are all functions
of x.
First take logs of base e
w
uv
y
w
v
u
y ln
ln
ln
ln
dx
dw
w
dx
dv
v
dx
du
u
dx
dy
y
1
1
1
1
dx
dw
w
dx
dv
v
dx
du
u
y
dx
dy 1
1
1
10. 10
Example 4: find dy/dx for the following.
Solution :
x
x
e
y
x
e
x
y
x
x
x
y
x
x
2
cosh
.
3
tan
.
2
2
cos
sin
.
1 3
4
3
4
2
x
x
x
x
x
x
dx
dy
x
x
x
x
x
x
x
x
x
dx
dy
dx
dw
w
dx
dv
v
dx
du
u
y
dx
dy
x
w
x
v
x
u
let
2
tan
2
cot
2
2
cos
sin
2
cos
)
sin
2
(
sin
cos
2
2
cos
sin
1
1
1
2
cos
,
sin
,
.
1
2
2
2
2
x
x
x
x
e
x
dx
dy
x
x
e
e
x
x
x
e
x
dx
dy
dx
dw
w
dx
dv
v
dx
du
u
y
dx
dy
x
w
e
v
x
u
let
x
x
x
x
x
tan
sec
3
4
tan
tan
sec
3
4
tan
1
1
1
tan
,
,
.
2
2
3
4
2
3
3
4
3
3
4
3
4
x
x
x
x
e
x
x
x
x
e
e
x
x
e
dx
dy
dx
dw
w
dx
dv
v
dx
du
u
y
dx
dy
x
w
x
v
e
u
let
x
x
x
x
x
2
tanh
2
3
4
2
cos
2
cosh
2
sinh
2
3
4
2
cos
1
1
1
2
cosh
,
,
.
3
3
4
3
2
4
4
3
4
3
4
11. 11
Parametric equations
In same cases, it is convenient to represent a function by
expressing x and y separately in terms of a third independent
variable. E.g. y= cos2t, x=sint.
The third variable is called parameter, and the two expression
of x and y are called parametric equation.
Example 5: find and for the following.
Solution :
1.
t
t
y
t
t
x
x
y
t
x
t
y
1
2
3
,
1
3
2
.
3
cos
,
sin
sin
3
.
2
sin
,
2
cos
.
1 3
3
dx
dy
2
2
dx
y
d
t
dt
dx
t
dt
dy
cos
2
sin
2
dx
dt
dt
dy
dt
dt
dx
dy
dx
dy
.
.
t
t
dx
dy
cos
1
.
2
sin
2
15. Exponential and logarithmic functions
• Today’s Objective:
When you have completed this lecture you will be able to:
• Solve indicial and logarithmic equations.
• Recognize that the exponential function and the natural
logarithmic function are mutual inverses
• Construct the hyperbolic functions from the odd and
even parts of the exponential function.
• Application of logarithms and exponential function.
15
16. Introduction to logarithms
If a number y can be written in the form , then the
index x is called the ‘logarithm of y to the base of a’,
Logarithms having a base of 10 are called common
logarithms and is usually abbreviated to .
16
10
log
x
a
y
y
x
then
a
y
if a
x
log
lg
Laws of logarithms
There are three laws of logarithms, which apply to any base:
B
A
A
n
A
B
A
B
A
B
A
n
log
log
)
log(
.
3
log
)
log(
.
2
log
log
)
log(
.
1
17. An indicial equation is an equation where the variable appears as an
index and the solution of such an equation requires the application of
logarithms.
17
Indicial equations
Example 1:
Find the value of X, give that 4
.
35
122
x
Solution :
7177
.
0
0792
.
1
2
5490
.
1
5490
.
1
0792
.
1
)
2
(
4
.
35
log
12
log
)
2
(
4
.
35
log
12
log 2
x
x
x
x
A
n
An
log
)
log(
18. 18
Example 2:
Find the value of X, give that
Solution :
694
.
6
3913
.
0
6192
.
2
2042
.
1
4150
.
1
)
4150
.
1
8063
.
1
(
4150
.
1
4150
.
1
2042
.
1
8063
.
1
4150
.
1
)
1
(
6021
.
0
)
2
3
(
26
log
)
1
(
4
log
)
2
3
(
26
log
4
log 1
2
3
x
x
x
x
x
x
x
x
x
x
log(An
)= nlogA
1
2
3
26
4
x
x
19. 19
Example 3:
Find the value of X, give that
Solution :
4853
.
2
5164
.
0
2834
.
1
2834
.
1
)
0436
.
2
5600
.
2
(
0436
.
2
2834
.
1
5600
.
2
0436
.
2
9138
.
0
8276
.
1
19726
.
2
7324
.
0
0436
.
2
9138
.
0
)
1
2
(
7324
.
0
)
3
(
)
6812
.
0
(
3
2
.
8
log
)
1
2
(
4
.
5
log
)
3
(
8
.
4
log
3
2
.
8
log
4
.
5
log
8
.
4
log
)
2
.
8
4
.
5
log(
1
2
3
3
1
2
3
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
B
A
B
A log
log
)
log(
x
x
x 3
1
2
3
8
.
4
2
.
8
4
.
5
20. 20
Example 4: Find the value of X, give that
Solution :
5006
.
1
7676
.
2
1533
.
4
1533
.
4
7676
.
2
4683
.
2
6123
.
1
7766
.
5
1553
.
1
8450
.
0
294
log
4
.
6
log
2
3
.
14
log
)
5
(
7
log
294
log
}
4
.
6
)
3
.
14
(
7
log{ 2
5
x
x
x
x
x
x
x
x
294
4
.
6
)
3
.
14
(
7 2
5
x
x
Example 5: Solve the equation
Solution :
2
1
4
2
2
2
4
0
)
4
(
2
0
)
2
(
0
)
4
)(
2
(
0
8
6
2
2
x
x
z
or
z
z
z
or
z
z
z
z
z
z
z
let
x
x
x
0
8
2
6
22
x
x
21. Graphs of logarithmic functions
21
A graph of is shown in bellow
x
y 10
log
0
1
log
a
0
loga
1
log
a
a
In general, with a logarithm
to any base a, it is noted
that:
22. Exponential functions
The exponential function is expressed by the equation:
or
where e is the exponential number 2.7182818 _ _ _ . The
graph of this function lies entirely above the x-axis as does
the graph of its reciprocal , as can be seen in the
diagram:
22
)
exp(x
y
x
e
y
x
e
y
23. The value of can be found to any level of precision desired
from the series expansion:
In practice a calculator is used.
Logarithms with base exponential called Napierian logarithms which is
written as:
So,
23
x
e
x
x
e ln
log
!
5
!
4
!
3
!
2
1
5
4
3
2
x
x
x
x
x
ex
x
e
x
e x
x
e
ln
log
.
..........
ln 3
x
e
24. 24
Example 6: Solve for X, given that
Solution :
)
1
(
60
20 2
x
e
810
.
0
405
.
0
2
405
.
0
2
2
3
ln
ln
2
3
3
2
6
2
1
)
1
(
60
20
2
2
2
2
2
x
x
x
e
e
e
e
e
x
x
x
x
x
Example 7: Solve for X, given that
Solution :
0
2
3
2
x
x
e
e
0
)
2
)(
1
(
0
2
3
2
y
y
y
y
e
y
let x
0
1
ln
1
ln
ln
1
1
0
)
1
(
x
x
e
e
y
y
x
x
693
.
0
2
ln
2
ln
ln
2
2
0
)
2
(
x
x
e
e
y
y
x
x
25. 25
Example 8:
Solution : Ans. 0.5545s
The current amperes flowing in a capacitor at time seconds is
given by
where and
Determine :
(a) the time for the current to reach 6.0 A.
(b) Sketch the graph of current against time.
)
1
(
8 CR
t
e
i
i t
3
10
25
R f
C 6
10
16
t
26. 26
Odd and even parts
Not every function is either even or odd but many can be written as
the sum of an even part and an odd part. If, given f(x) where f( -x) is
also defined then:
is even and is odd
Furthermore is called the even part of f(x) and is called
the odd part of f(x).
For example, if then
So that the even and odd parts of are:
2
)
(
)
(
)
(
x
f
x
f
x
fe
2
)
(
)
(
)
(
x
f
x
f
x
fo
)
(x
fe )
(x
fo
1
2
3
)
( 2
x
x
x
f 1
2
3
1
)
(
2
)
(
3
)
( 2
2
x
x
x
x
x
f
)
(x
f
x
x
x
x
x
x
f
x
x
x
x
x
x
f
o
e
2
2
)
1
2
3
(
)
1
2
3
(
)
(
1
3
2
)
1
2
3
(
)
1
2
3
(
)
(
2
2
2
2
2
So, the even and odd parts of the odd part of
are………………
4
3
2
)
( 2
3
x
x
x
x
f
27. 27
Odd and even parts of the exponential function
The exponential function is neither odd nor even but it can be written
as a sum of an odd part and an even part.
That is, and . These
two functions are known as the hyperbolic cosine and the hyperbolic
sine respectively:
and
Using these two functions the hyperbolic tangent can also be defined:
The logarithmic function is neither odd nor even and indeed
does not possess even and odd parts because is not defined.
2
)
exp(
)
exp(
)
(
exp
x
x
x
e
2
)
exp(
)
exp(
)
(
exp
x
x
x
o
2
cosh
x
x
e
e
x
2
sinh
x
x
e
e
x
x
x
x
x
e
e
e
e
x
tanh
x
y a
log
)
(
log x
a
