ICAI Notes- Differential Calculus | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


BASIC CONCEPTS OF DIFFERENTIAL
AND INTEGRAL CALCULUS
8
CHAPTER
Differential Calculus
Basic Laws of
Differentiation
Some Standard
Results
C al c ulu s
After reading this chapter, students will be able to understand:
? Understand the basics of differentiation and integration.
? Know how to compute derivative of a function by the first principle, derivative of a function
by the application of formulae and higher order differentiation.
? Appreciate various techniques of integration.
? Understand the concept of definite of integrals of functions and its application.
CHAPTER OVERVIEW
Integral Calculus
Methods of
Substitution
Basic
Formulas
© The Institute of Chartered Accountants of India
Page 2


BASIC CONCEPTS OF DIFFERENTIAL
AND INTEGRAL CALCULUS
8
CHAPTER
Differential Calculus
Basic Laws of
Differentiation
Some Standard
Results
C al c ulu s
After reading this chapter, students will be able to understand:
? Understand the basics of differentiation and integration.
? Know how to compute derivative of a function by the first principle, derivative of a function
by the application of formulae and higher order differentiation.
? Appreciate various techniques of integration.
? Understand the concept of definite of integrals of functions and its application.
CHAPTER OVERVIEW
Integral Calculus
Methods of
Substitution
Basic
Formulas
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
8. 2
Differentiation is one of the most important fundamental operations in calculus. Its theory
primarily depends on the idea of limit and continuity of function.
To express the rate of change in any function we introduce concept of derivative which
involves a very small change in the dependent variable with reference to a very small change
in independent variable.
Thus differentiation is the process of finding the derivative of a continuous function. It is
defined as the limiting value of the ratio of the change (increment) in the function corresponding
to a small change (increment) in the independent variable (argument) as the later tends to
zero.
Let y = f(x) be a function. If h (or ?x) be the small increment in x and the corresponding
increment in y or f(x) be ?y = f(x+h) – f(x) then the derivative of f(x) is defined
as 
? h0
f(x+h) - f(x)
lim
h
 i.e.
     = 
x
) x (f ) x x (f
lim
0x
?
? ? ?
? ?
This is denoted as f’(x) or dy/ dx or 
d
dx
f(x). The derivative of f(x) is also known as differential
coefficient of f(x) with respect to x. This process of differentiation is called the first principle
(or definition or abinitio) (Ab-initio).
Note: In the light of above discussion a function f (x) is said to differentiable at x = c if
hc
f(x)-f(c)
lim
x-c
?
 exist which is called the differential coefficient of f(x) at x = c and is denoted
by f
 ‘
(c) or
c x
dx
dy
?
?
?
?
?
?
?
.
We will now study this with an example.
Consider the function f(x) = x
2
.
© The Institute of Chartered Accountants of India
Page 3


BASIC CONCEPTS OF DIFFERENTIAL
AND INTEGRAL CALCULUS
8
CHAPTER
Differential Calculus
Basic Laws of
Differentiation
Some Standard
Results
C al c ulu s
After reading this chapter, students will be able to understand:
? Understand the basics of differentiation and integration.
? Know how to compute derivative of a function by the first principle, derivative of a function
by the application of formulae and higher order differentiation.
? Appreciate various techniques of integration.
? Understand the concept of definite of integrals of functions and its application.
CHAPTER OVERVIEW
Integral Calculus
Methods of
Substitution
Basic
Formulas
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
8. 2
Differentiation is one of the most important fundamental operations in calculus. Its theory
primarily depends on the idea of limit and continuity of function.
To express the rate of change in any function we introduce concept of derivative which
involves a very small change in the dependent variable with reference to a very small change
in independent variable.
Thus differentiation is the process of finding the derivative of a continuous function. It is
defined as the limiting value of the ratio of the change (increment) in the function corresponding
to a small change (increment) in the independent variable (argument) as the later tends to
zero.
Let y = f(x) be a function. If h (or ?x) be the small increment in x and the corresponding
increment in y or f(x) be ?y = f(x+h) – f(x) then the derivative of f(x) is defined
as 
? h0
f(x+h) - f(x)
lim
h
 i.e.
     = 
x
) x (f ) x x (f
lim
0x
?
? ? ?
? ?
This is denoted as f’(x) or dy/ dx or 
d
dx
f(x). The derivative of f(x) is also known as differential
coefficient of f(x) with respect to x. This process of differentiation is called the first principle
(or definition or abinitio) (Ab-initio).
Note: In the light of above discussion a function f (x) is said to differentiable at x = c if
hc
f(x)-f(c)
lim
x-c
?
 exist which is called the differential coefficient of f(x) at x = c and is denoted
by f
 ‘
(c) or
c x
dx
dy
?
?
?
?
?
?
?
.
We will now study this with an example.
Consider the function f(x) = x
2
.
© The Institute of Chartered Accountants of India
8 . 3 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS
By definition
x
x ) x ( x x 2 x
lim
x
x ) x x (
lim
x
) x ( f ) x x ( f
lim ) x ( f
dx
d
2 2 2
0 x
2 2
0 x 0 x
?
? ? ? ? ?
?
?
? ? ?
?
?
? ? ?
?
? ? ? ? ? ?
  = 
x 2 0 x 2 ) x x 2 ( lim
0 x
? ? ? ? ?
? ?
Thus, derivative of f(x) exists for all values of x and equals 2x at any point x.
Examples of differentiations from the 1st principle
i) f(x) = c, c being a constant.
Since c is constant we may write f(x+h) = c.
So f(x+h) – f(x) = 0
Hence 
h 0 h 0
f(x+h)- f(x) 0
f'(x)=lim =lim
h h
? ?
 = 0
So 
d(c)
dx
= 0
ii) Let f(x) = x
n
; then f(x+h) = (x+h)
n
let x+h =t or h= t – x and as h ?0, t ?x
Now f’(x) = 
h 0
lim
?
f(x+h)-f(x)
h
= 
h 0
lim
?
n n
(x+h) -x
h
= 
t x
lim
?
 (t
n 
– x
n
 ) / (t – x) = nx 
n–1
Hence 
n
d
(x )
dx
 = nx 
n–1
iii) f (x) = e
x 
?
 f(x + h) = e 
x+h
So f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
= 
h 0
lim
?
x+h x
e - e
h
 = 
h 0
lim
?
x h
e (e -1)
h
= e
x 
h 0
lim
?
h
e -1
h
= e
x
.1
Hence 
d
dx
 (e
x
) = e
x
© The Institute of Chartered Accountants of India
Page 4


BASIC CONCEPTS OF DIFFERENTIAL
AND INTEGRAL CALCULUS
8
CHAPTER
Differential Calculus
Basic Laws of
Differentiation
Some Standard
Results
C al c ulu s
After reading this chapter, students will be able to understand:
? Understand the basics of differentiation and integration.
? Know how to compute derivative of a function by the first principle, derivative of a function
by the application of formulae and higher order differentiation.
? Appreciate various techniques of integration.
? Understand the concept of definite of integrals of functions and its application.
CHAPTER OVERVIEW
Integral Calculus
Methods of
Substitution
Basic
Formulas
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
8. 2
Differentiation is one of the most important fundamental operations in calculus. Its theory
primarily depends on the idea of limit and continuity of function.
To express the rate of change in any function we introduce concept of derivative which
involves a very small change in the dependent variable with reference to a very small change
in independent variable.
Thus differentiation is the process of finding the derivative of a continuous function. It is
defined as the limiting value of the ratio of the change (increment) in the function corresponding
to a small change (increment) in the independent variable (argument) as the later tends to
zero.
Let y = f(x) be a function. If h (or ?x) be the small increment in x and the corresponding
increment in y or f(x) be ?y = f(x+h) – f(x) then the derivative of f(x) is defined
as 
? h0
f(x+h) - f(x)
lim
h
 i.e.
     = 
x
) x (f ) x x (f
lim
0x
?
? ? ?
? ?
This is denoted as f’(x) or dy/ dx or 
d
dx
f(x). The derivative of f(x) is also known as differential
coefficient of f(x) with respect to x. This process of differentiation is called the first principle
(or definition or abinitio) (Ab-initio).
Note: In the light of above discussion a function f (x) is said to differentiable at x = c if
hc
f(x)-f(c)
lim
x-c
?
 exist which is called the differential coefficient of f(x) at x = c and is denoted
by f
 ‘
(c) or
c x
dx
dy
?
?
?
?
?
?
?
.
We will now study this with an example.
Consider the function f(x) = x
2
.
© The Institute of Chartered Accountants of India
8 . 3 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS
By definition
x
x ) x ( x x 2 x
lim
x
x ) x x (
lim
x
) x ( f ) x x ( f
lim ) x ( f
dx
d
2 2 2
0 x
2 2
0 x 0 x
?
? ? ? ? ?
?
?
? ? ?
?
?
? ? ?
?
? ? ? ? ? ?
  = 
x 2 0 x 2 ) x x 2 ( lim
0 x
? ? ? ? ?
? ?
Thus, derivative of f(x) exists for all values of x and equals 2x at any point x.
Examples of differentiations from the 1st principle
i) f(x) = c, c being a constant.
Since c is constant we may write f(x+h) = c.
So f(x+h) – f(x) = 0
Hence 
h 0 h 0
f(x+h)- f(x) 0
f'(x)=lim =lim
h h
? ?
 = 0
So 
d(c)
dx
= 0
ii) Let f(x) = x
n
; then f(x+h) = (x+h)
n
let x+h =t or h= t – x and as h ?0, t ?x
Now f’(x) = 
h 0
lim
?
f(x+h)-f(x)
h
= 
h 0
lim
?
n n
(x+h) -x
h
= 
t x
lim
?
 (t
n 
– x
n
 ) / (t – x) = nx 
n–1
Hence 
n
d
(x )
dx
 = nx 
n–1
iii) f (x) = e
x 
?
 f(x + h) = e 
x+h
So f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
= 
h 0
lim
?
x+h x
e - e
h
 = 
h 0
lim
?
x h
e (e -1)
h
= e
x 
h 0
lim
?
h
e -1
h
= e
x
.1
Hence 
d
dx
 (e
x
) = e
x
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
8 . 4
iv) Let f(x) = a
x
 then f(x+h) = a
x+h
f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
 = 
h 0
lim
?
x+h x
a -a
h
 = 
h 0
lim
?
?
?
?
?
?
? ?
h
) 1 a ( a
h x
= a
x
h 0
lim
?
h
a -1
h
= a
x
 log
e
a
Thus 
d
dx
 (a
x
) = a
x
 log
e
a
v) Let f(x) = 
x
. Then f(x + h) = 
x+h
f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
= 
h 0
lim
?
x+h - x
h
= 
h 0
lim
?
( x+h - x ) ( x+h + x)
h( x+h + x)
= 
h 0
lim
?
x+h-x
h ( x+h+ x
= 
h 0
lim
?
1 1
=
x+h + x 2 x
Thus 
d
( x)
dx
 = 
1
2 x
vi) f(x) = log x 
?
 f(x + h) = log ( x + h)
f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
= 
h 0
lim
?
log (x+h)- logx
h
= 
0 h
lim
?
 
h
x
h x
log
?
?
?
?
?
?
?
= 
0 h
lim
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x
h
1 log
h
1
© The Institute of Chartered Accountants of India
Page 5


BASIC CONCEPTS OF DIFFERENTIAL
AND INTEGRAL CALCULUS
8
CHAPTER
Differential Calculus
Basic Laws of
Differentiation
Some Standard
Results
C al c ulu s
After reading this chapter, students will be able to understand:
? Understand the basics of differentiation and integration.
? Know how to compute derivative of a function by the first principle, derivative of a function
by the application of formulae and higher order differentiation.
? Appreciate various techniques of integration.
? Understand the concept of definite of integrals of functions and its application.
CHAPTER OVERVIEW
Integral Calculus
Methods of
Substitution
Basic
Formulas
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
8. 2
Differentiation is one of the most important fundamental operations in calculus. Its theory
primarily depends on the idea of limit and continuity of function.
To express the rate of change in any function we introduce concept of derivative which
involves a very small change in the dependent variable with reference to a very small change
in independent variable.
Thus differentiation is the process of finding the derivative of a continuous function. It is
defined as the limiting value of the ratio of the change (increment) in the function corresponding
to a small change (increment) in the independent variable (argument) as the later tends to
zero.
Let y = f(x) be a function. If h (or ?x) be the small increment in x and the corresponding
increment in y or f(x) be ?y = f(x+h) – f(x) then the derivative of f(x) is defined
as 
? h0
f(x+h) - f(x)
lim
h
 i.e.
     = 
x
) x (f ) x x (f
lim
0x
?
? ? ?
? ?
This is denoted as f’(x) or dy/ dx or 
d
dx
f(x). The derivative of f(x) is also known as differential
coefficient of f(x) with respect to x. This process of differentiation is called the first principle
(or definition or abinitio) (Ab-initio).
Note: In the light of above discussion a function f (x) is said to differentiable at x = c if
hc
f(x)-f(c)
lim
x-c
?
 exist which is called the differential coefficient of f(x) at x = c and is denoted
by f
 ‘
(c) or
c x
dx
dy
?
?
?
?
?
?
?
.
We will now study this with an example.
Consider the function f(x) = x
2
.
© The Institute of Chartered Accountants of India
8 . 3 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS
By definition
x
x ) x ( x x 2 x
lim
x
x ) x x (
lim
x
) x ( f ) x x ( f
lim ) x ( f
dx
d
2 2 2
0 x
2 2
0 x 0 x
?
? ? ? ? ?
?
?
? ? ?
?
?
? ? ?
?
? ? ? ? ? ?
  = 
x 2 0 x 2 ) x x 2 ( lim
0 x
? ? ? ? ?
? ?
Thus, derivative of f(x) exists for all values of x and equals 2x at any point x.
Examples of differentiations from the 1st principle
i) f(x) = c, c being a constant.
Since c is constant we may write f(x+h) = c.
So f(x+h) – f(x) = 0
Hence 
h 0 h 0
f(x+h)- f(x) 0
f'(x)=lim =lim
h h
? ?
 = 0
So 
d(c)
dx
= 0
ii) Let f(x) = x
n
; then f(x+h) = (x+h)
n
let x+h =t or h= t – x and as h ?0, t ?x
Now f’(x) = 
h 0
lim
?
f(x+h)-f(x)
h
= 
h 0
lim
?
n n
(x+h) -x
h
= 
t x
lim
?
 (t
n 
– x
n
 ) / (t – x) = nx 
n–1
Hence 
n
d
(x )
dx
 = nx 
n–1
iii) f (x) = e
x 
?
 f(x + h) = e 
x+h
So f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
= 
h 0
lim
?
x+h x
e - e
h
 = 
h 0
lim
?
x h
e (e -1)
h
= e
x 
h 0
lim
?
h
e -1
h
= e
x
.1
Hence 
d
dx
 (e
x
) = e
x
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
8 . 4
iv) Let f(x) = a
x
 then f(x+h) = a
x+h
f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
 = 
h 0
lim
?
x+h x
a -a
h
 = 
h 0
lim
?
?
?
?
?
?
? ?
h
) 1 a ( a
h x
= a
x
h 0
lim
?
h
a -1
h
= a
x
 log
e
a
Thus 
d
dx
 (a
x
) = a
x
 log
e
a
v) Let f(x) = 
x
. Then f(x + h) = 
x+h
f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
= 
h 0
lim
?
x+h - x
h
= 
h 0
lim
?
( x+h - x ) ( x+h + x)
h( x+h + x)
= 
h 0
lim
?
x+h-x
h ( x+h+ x
= 
h 0
lim
?
1 1
=
x+h + x 2 x
Thus 
d
( x)
dx
 = 
1
2 x
vi) f(x) = log x 
?
 f(x + h) = log ( x + h)
f’(x) = 
h 0
lim
?
f(x+h)- f(x)
h
= 
h 0
lim
?
log (x+h)- logx
h
= 
0 h
lim
?
 
h
x
h x
log
?
?
?
?
?
?
?
= 
0 h
lim
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x
h
1 log
h
1
© The Institute of Chartered Accountants of India
8. 5 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS
Let
h
x
 = t i.e. h=tx and as h ? 0,  t ? 0
?
f’(x) = 
t0
lim
?
11
log(1+t)= 
tx x t0
lim
?
1 11
log(1+t) = ×1=
t xx
, since 
t0
lim
?
? ?
log 1 + t
1
t
?
Thus 
d
dx
 (log x) = 
1
x
(1)
d
dx
 (x
n
) = nx 
n–1
(2) 
d
dx
 (e
x
) = e
x
(3) 
d
dx
 (a
x
) = a
x 
log 
e
 a
(4)
d
dx
 (constant) = 0 (5) 
d
dx
 (e
ax
) = ae 
ax
(5) 
d
dx
 (log x) = 
1
x
Note:
d
dx
 { c f(x)} = cf’(x) c being constant.
In brief we may write below the above functions and their derivatives:
Table: Few functions and their derivatives
Function derivative of the function
f(x) f
 ‘
(x)
x
 n
n x
 n – 1
e
a x
ae
 a x
log x 1/ x
a
 x
a
 x
 log
 e
a
c (a constant) 0
We also tabulate the basic laws of differentiation.
© The Institute of Chartered Accountants of India