Page 1
DIFFERENTIABILITY
Differentiability is a fundamental concept in calculus that extends the idea of continuity. It focuses on
how smoothly a function changes at a given point, providing insight into the function's rate of change
and local behavior. This concept is crucial for analyzing functions, solving optimization problems, and
applying calculus across various fields of science and engineering.
This section delves into the meaning of derivatives, explores the criteria for differentiability, and
examines its relationship with continuity. Through a combination of theoretical explanations and
practical examples, we'll investigate the nuances of differentiability, including special cases and
geometric interpretations. By understanding differentiability, we gain powerful tools for mathematical
analysis and problemsolving in calculus.
1. MEANING OF DERIVATIVE:
The instantaneous rate of change of a function with respect to the dependent variable is called derivative.
Let ' ?? ' be a given function of one variable and let ?? ?? denote a number (positive or negative) to be added
to the number ?? . Let ?? ?? denote the corresponding change of ' ?? ' then ?? ?? = ?? (?? + ???? )  ?? (?? )
?
????
????
=
?? (?? + ???? )  ?? (?? )
????
If ???? /???? approaches a limit as ???? approaches zero, this limit is the derivative of ' ?? ' at the point ?? . The
derivative of a function ' ?? ' is a function ; this function is denoted by symbols such as
?? '
(?? ),
????
????
,
?? ????
?? (?? ) ????
???? (?? )
????
?
????
????
= ?????? ???? ?0
?
????
????
= ?????? ???? ?0
?
?? (?? + ???? )  ?? (?? )
????
The derivative evaluated at a point ?? , is written, ?? '
(?? ),
???? (?? )
????

?? =?? , ?? '
(?? )
?? =?? , etc.
2. EXISTENCE OF DERIVATIVE AT x= a
(a) Right hand derivative:
The right hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
Page 2
DIFFERENTIABILITY
Differentiability is a fundamental concept in calculus that extends the idea of continuity. It focuses on
how smoothly a function changes at a given point, providing insight into the function's rate of change
and local behavior. This concept is crucial for analyzing functions, solving optimization problems, and
applying calculus across various fields of science and engineering.
This section delves into the meaning of derivatives, explores the criteria for differentiability, and
examines its relationship with continuity. Through a combination of theoretical explanations and
practical examples, we'll investigate the nuances of differentiability, including special cases and
geometric interpretations. By understanding differentiability, we gain powerful tools for mathematical
analysis and problemsolving in calculus.
1. MEANING OF DERIVATIVE:
The instantaneous rate of change of a function with respect to the dependent variable is called derivative.
Let ' ?? ' be a given function of one variable and let ?? ?? denote a number (positive or negative) to be added
to the number ?? . Let ?? ?? denote the corresponding change of ' ?? ' then ?? ?? = ?? (?? + ???? )  ?? (?? )
?
????
????
=
?? (?? + ???? )  ?? (?? )
????
If ???? /???? approaches a limit as ???? approaches zero, this limit is the derivative of ' ?? ' at the point ?? . The
derivative of a function ' ?? ' is a function ; this function is denoted by symbols such as
?? '
(?? ),
????
????
,
?? ????
?? (?? ) ????
???? (?? )
????
?
????
????
= ?????? ???? ?0
?
????
????
= ?????? ???? ?0
?
?? (?? + ???? )  ?? (?? )
????
The derivative evaluated at a point ?? , is written, ?? '
(?? ),
???? (?? )
????

?? =?? , ?? '
(?? )
?? =?? , etc.
2. EXISTENCE OF DERIVATIVE AT x= a
(a) Right hand derivative:
The right hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
?? ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
, provided the limit exists & is finite. ( h > 0 )
(b) Left hand derivative:
The left hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
????
'
(?? ) =
h?0
?? (?? h)?? (?? )
h
, provided the limit exists & is finite. (h > 0 )
Hence ?? (?? ) is said to be derivable or differentiable at ?? = ?? . If ????
'
(?? ) = ????
'
(?? ) = finite quantity
and it is denoted by ?? (?? ); where ?? (?? ) = ????
'
(?? ) = ????
'
(a) & it is called derivative or differential
coefficient of ?? (?? ) at ?? = ?? .
3. DIFFERENTIABILITY & CONTINUITY:
Theorem: If a function ?? (?? ) is derivable at ?? = ?? , then ?? (?? ) is continuous at ?? = ?? .
Proof: ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
exists.
Also ?? (?? + h)  ?? (?? ) =
?? (?? +h)?? (?? )
h
· h [h ? 0]
? ?????? h?0
[?? (?? + h)  ?? (?? )] = ?????? h?0
?? (?? + h)  ?? (?? )
h
· h = ?? '
(?? ) · 0 = 0
? ?????? h?0
[?? (?? + h)  ?? (?? )] = 0 ? ?????? h?0
?? (?? + h) = ?? (?? ) ? ?? (?? ) is continuous at ?? = ?? .
Note:
(i) ???????????????????????????? ? ???????????????????? ; ???????????????????? ?? ???????????????????????????? ; ?????? ???????????????????????????? ?
? ?????? ???????????????????? ?????? ?????? ???????????????????? ? ?????? ????????????????????????????
(ii) All polynomial, rational, trigonometric, logarithmic and exponential function are continuous and
differentiable in their domains.
(iii) If ?? (?? )&?? (?? ) are differentiable at ?? = a, then the functions ?? (?? ) + ?? (?? ), ?? (?? )  ?? (?? ), ?? (?? ). ?? (?? ) will
also be differentiable at ?? = ?? & if ?? (?? ) ? 0 then the function ?? (?? )/?? (?? ) will also be differentiable at ?? =
?? .
?? = 0&
?? 2
Solution:
?? (?? ) = {1 + ?? ; 8 < ?? < 0  1 + ?????? ?? ; 0 = ?? <
?? 2
?????? ?? ;
?? 2
= ?? < 8
To check the differentiability at ?? = 0
?????? = ?????? h?0
?
?? (0  h)  ?? (0)
h
= ?????? h?0
?
1 + 0  h  (1)
h
= 1
Page 3
DIFFERENTIABILITY
Differentiability is a fundamental concept in calculus that extends the idea of continuity. It focuses on
how smoothly a function changes at a given point, providing insight into the function's rate of change
and local behavior. This concept is crucial for analyzing functions, solving optimization problems, and
applying calculus across various fields of science and engineering.
This section delves into the meaning of derivatives, explores the criteria for differentiability, and
examines its relationship with continuity. Through a combination of theoretical explanations and
practical examples, we'll investigate the nuances of differentiability, including special cases and
geometric interpretations. By understanding differentiability, we gain powerful tools for mathematical
analysis and problemsolving in calculus.
1. MEANING OF DERIVATIVE:
The instantaneous rate of change of a function with respect to the dependent variable is called derivative.
Let ' ?? ' be a given function of one variable and let ?? ?? denote a number (positive or negative) to be added
to the number ?? . Let ?? ?? denote the corresponding change of ' ?? ' then ?? ?? = ?? (?? + ???? )  ?? (?? )
?
????
????
=
?? (?? + ???? )  ?? (?? )
????
If ???? /???? approaches a limit as ???? approaches zero, this limit is the derivative of ' ?? ' at the point ?? . The
derivative of a function ' ?? ' is a function ; this function is denoted by symbols such as
?? '
(?? ),
????
????
,
?? ????
?? (?? ) ????
???? (?? )
????
?
????
????
= ?????? ???? ?0
?
????
????
= ?????? ???? ?0
?
?? (?? + ???? )  ?? (?? )
????
The derivative evaluated at a point ?? , is written, ?? '
(?? ),
???? (?? )
????

?? =?? , ?? '
(?? )
?? =?? , etc.
2. EXISTENCE OF DERIVATIVE AT x= a
(a) Right hand derivative:
The right hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
?? ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
, provided the limit exists & is finite. ( h > 0 )
(b) Left hand derivative:
The left hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
????
'
(?? ) =
h?0
?? (?? h)?? (?? )
h
, provided the limit exists & is finite. (h > 0 )
Hence ?? (?? ) is said to be derivable or differentiable at ?? = ?? . If ????
'
(?? ) = ????
'
(?? ) = finite quantity
and it is denoted by ?? (?? ); where ?? (?? ) = ????
'
(?? ) = ????
'
(a) & it is called derivative or differential
coefficient of ?? (?? ) at ?? = ?? .
3. DIFFERENTIABILITY & CONTINUITY:
Theorem: If a function ?? (?? ) is derivable at ?? = ?? , then ?? (?? ) is continuous at ?? = ?? .
Proof: ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
exists.
Also ?? (?? + h)  ?? (?? ) =
?? (?? +h)?? (?? )
h
· h [h ? 0]
? ?????? h?0
[?? (?? + h)  ?? (?? )] = ?????? h?0
?? (?? + h)  ?? (?? )
h
· h = ?? '
(?? ) · 0 = 0
? ?????? h?0
[?? (?? + h)  ?? (?? )] = 0 ? ?????? h?0
?? (?? + h) = ?? (?? ) ? ?? (?? ) is continuous at ?? = ?? .
Note:
(i) ???????????????????????????? ? ???????????????????? ; ???????????????????? ?? ???????????????????????????? ; ?????? ???????????????????????????? ?
? ?????? ???????????????????? ?????? ?????? ???????????????????? ? ?????? ????????????????????????????
(ii) All polynomial, rational, trigonometric, logarithmic and exponential function are continuous and
differentiable in their domains.
(iii) If ?? (?? )&?? (?? ) are differentiable at ?? = a, then the functions ?? (?? ) + ?? (?? ), ?? (?? )  ?? (?? ), ?? (?? ). ?? (?? ) will
also be differentiable at ?? = ?? & if ?? (?? ) ? 0 then the function ?? (?? )/?? (?? ) will also be differentiable at ?? =
?? .
?? = 0&
?? 2
Solution:
?? (?? ) = {1 + ?? ; 8 < ?? < 0  1 + ?????? ?? ; 0 = ?? <
?? 2
?????? ?? ;
?? 2
= ?? < 8
To check the differentiability at ?? = 0
?????? = ?????? h?0
?
?? (0  h)  ?? (0)
h
= ?????? h?0
?
1 + 0  h  (1)
h
= 1
?????? = ?????? h?0
?
?? (0 + h)  ?? (0)
h
= ?????? h?0
?
1 + ?????? h + 1
h
= 1 ? ?????? = ?????? ? ???????????????????????????? ???? ?? = 0.
? ???????????????????? ???? ?? = 0.
To check the continuity at ?? =
?? 2
LHL ?????? ?? ?
?? 2

??? (?? ) = ?????? ?? ?
?? 
2
?(1 + ?????? ?? ) = 0
RHL ?????? ?? ?
?? +
2
??? (?? ) = ?????? ?? ?
?? +
2
??????? ?? = 0
? LHL = RHL = ?? (
?? 2
) = 0
? Continuous at ?? =
?? 2
.
To check the differentiability at ?? =
?? 2
?????? = ?????? h?0
?
?? (
?? 2
 h)  ?? (
?? 2
)
h
= ?????? h?0
?
1 + ?????? h  0
h
= 0 ?????? = ?????? h?0
?
?? (
?? 2
+ h)  ?? (
?? 2
)
h
= ?????? h?0
?
?????? h  0
h
= 1 ? ?????? ? ?????? ? ?????? ???????????????????????????? ???? ?? =
?? 2
.
Problem 2: If ?? (?? ) = {?? + ?? ?? 2
?? < 1 3????  ?? + 2?? = 1
then find ?? and ?? so that ?? (?? ) become differentiable at ?? = 1.
Solution:
????
'
(1) = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
3?? (1 + h)  ?? + 2  3?? + ??  2
h
= ?????? h?0
?
3?? h
h
= 3?? ?? ?? '
(1)
= ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
?? + ?? (1  h)
2
 3?? + ??  2
h
= ?????? h?0
?
(2?? + 2??  2) + ?? h
2
 2?? h
h
hence for this limit to be defined
2?? + 2??  2 = 0 ?? = ?? + 1 ?? ?? '
(1) = ?????? h?0
?  (?? h  2?? ) = 2?? ? ?? ?? '
(1) = ?? ?? '
(1) 3?? = 2?? = 2(?? + 1) ?? = 2, ?? = 3
Problem 3: ?? (?? ) = {[?????? ???? ] ?? = 1 2{?? }  1 ?? > 1 comment on the derivability at ?? = 1, where [ ]
denotes greatest integer function &{ }enotes fractional part function.
Solution:
?? ?? '
(1) = ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
[?????? (??  ?? h)] + 1
h
= ?????? h?0
?
1 + 1
h
= 0 ?? '
(1) = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
2{1 + h}  1 + 1
h
= ?????? h?0
?
2h
h
= 2
Hence ?? (?? ) is not differentiable at ?? = 1.
Page 4
DIFFERENTIABILITY
Differentiability is a fundamental concept in calculus that extends the idea of continuity. It focuses on
how smoothly a function changes at a given point, providing insight into the function's rate of change
and local behavior. This concept is crucial for analyzing functions, solving optimization problems, and
applying calculus across various fields of science and engineering.
This section delves into the meaning of derivatives, explores the criteria for differentiability, and
examines its relationship with continuity. Through a combination of theoretical explanations and
practical examples, we'll investigate the nuances of differentiability, including special cases and
geometric interpretations. By understanding differentiability, we gain powerful tools for mathematical
analysis and problemsolving in calculus.
1. MEANING OF DERIVATIVE:
The instantaneous rate of change of a function with respect to the dependent variable is called derivative.
Let ' ?? ' be a given function of one variable and let ?? ?? denote a number (positive or negative) to be added
to the number ?? . Let ?? ?? denote the corresponding change of ' ?? ' then ?? ?? = ?? (?? + ???? )  ?? (?? )
?
????
????
=
?? (?? + ???? )  ?? (?? )
????
If ???? /???? approaches a limit as ???? approaches zero, this limit is the derivative of ' ?? ' at the point ?? . The
derivative of a function ' ?? ' is a function ; this function is denoted by symbols such as
?? '
(?? ),
????
????
,
?? ????
?? (?? ) ????
???? (?? )
????
?
????
????
= ?????? ???? ?0
?
????
????
= ?????? ???? ?0
?
?? (?? + ???? )  ?? (?? )
????
The derivative evaluated at a point ?? , is written, ?? '
(?? ),
???? (?? )
????

?? =?? , ?? '
(?? )
?? =?? , etc.
2. EXISTENCE OF DERIVATIVE AT x= a
(a) Right hand derivative:
The right hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
?? ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
, provided the limit exists & is finite. ( h > 0 )
(b) Left hand derivative:
The left hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
????
'
(?? ) =
h?0
?? (?? h)?? (?? )
h
, provided the limit exists & is finite. (h > 0 )
Hence ?? (?? ) is said to be derivable or differentiable at ?? = ?? . If ????
'
(?? ) = ????
'
(?? ) = finite quantity
and it is denoted by ?? (?? ); where ?? (?? ) = ????
'
(?? ) = ????
'
(a) & it is called derivative or differential
coefficient of ?? (?? ) at ?? = ?? .
3. DIFFERENTIABILITY & CONTINUITY:
Theorem: If a function ?? (?? ) is derivable at ?? = ?? , then ?? (?? ) is continuous at ?? = ?? .
Proof: ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
exists.
Also ?? (?? + h)  ?? (?? ) =
?? (?? +h)?? (?? )
h
· h [h ? 0]
? ?????? h?0
[?? (?? + h)  ?? (?? )] = ?????? h?0
?? (?? + h)  ?? (?? )
h
· h = ?? '
(?? ) · 0 = 0
? ?????? h?0
[?? (?? + h)  ?? (?? )] = 0 ? ?????? h?0
?? (?? + h) = ?? (?? ) ? ?? (?? ) is continuous at ?? = ?? .
Note:
(i) ???????????????????????????? ? ???????????????????? ; ???????????????????? ?? ???????????????????????????? ; ?????? ???????????????????????????? ?
? ?????? ???????????????????? ?????? ?????? ???????????????????? ? ?????? ????????????????????????????
(ii) All polynomial, rational, trigonometric, logarithmic and exponential function are continuous and
differentiable in their domains.
(iii) If ?? (?? )&?? (?? ) are differentiable at ?? = a, then the functions ?? (?? ) + ?? (?? ), ?? (?? )  ?? (?? ), ?? (?? ). ?? (?? ) will
also be differentiable at ?? = ?? & if ?? (?? ) ? 0 then the function ?? (?? )/?? (?? ) will also be differentiable at ?? =
?? .
?? = 0&
?? 2
Solution:
?? (?? ) = {1 + ?? ; 8 < ?? < 0  1 + ?????? ?? ; 0 = ?? <
?? 2
?????? ?? ;
?? 2
= ?? < 8
To check the differentiability at ?? = 0
?????? = ?????? h?0
?
?? (0  h)  ?? (0)
h
= ?????? h?0
?
1 + 0  h  (1)
h
= 1
?????? = ?????? h?0
?
?? (0 + h)  ?? (0)
h
= ?????? h?0
?
1 + ?????? h + 1
h
= 1 ? ?????? = ?????? ? ???????????????????????????? ???? ?? = 0.
? ???????????????????? ???? ?? = 0.
To check the continuity at ?? =
?? 2
LHL ?????? ?? ?
?? 2

??? (?? ) = ?????? ?? ?
?? 
2
?(1 + ?????? ?? ) = 0
RHL ?????? ?? ?
?? +
2
??? (?? ) = ?????? ?? ?
?? +
2
??????? ?? = 0
? LHL = RHL = ?? (
?? 2
) = 0
? Continuous at ?? =
?? 2
.
To check the differentiability at ?? =
?? 2
?????? = ?????? h?0
?
?? (
?? 2
 h)  ?? (
?? 2
)
h
= ?????? h?0
?
1 + ?????? h  0
h
= 0 ?????? = ?????? h?0
?
?? (
?? 2
+ h)  ?? (
?? 2
)
h
= ?????? h?0
?
?????? h  0
h
= 1 ? ?????? ? ?????? ? ?????? ???????????????????????????? ???? ?? =
?? 2
.
Problem 2: If ?? (?? ) = {?? + ?? ?? 2
?? < 1 3????  ?? + 2?? = 1
then find ?? and ?? so that ?? (?? ) become differentiable at ?? = 1.
Solution:
????
'
(1) = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
3?? (1 + h)  ?? + 2  3?? + ??  2
h
= ?????? h?0
?
3?? h
h
= 3?? ?? ?? '
(1)
= ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
?? + ?? (1  h)
2
 3?? + ??  2
h
= ?????? h?0
?
(2?? + 2??  2) + ?? h
2
 2?? h
h
hence for this limit to be defined
2?? + 2??  2 = 0 ?? = ?? + 1 ?? ?? '
(1) = ?????? h?0
?  (?? h  2?? ) = 2?? ? ?? ?? '
(1) = ?? ?? '
(1) 3?? = 2?? = 2(?? + 1) ?? = 2, ?? = 3
Problem 3: ?? (?? ) = {[?????? ???? ] ?? = 1 2{?? }  1 ?? > 1 comment on the derivability at ?? = 1, where [ ]
denotes greatest integer function &{ }enotes fractional part function.
Solution:
?? ?? '
(1) = ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
[?????? (??  ?? h)] + 1
h
= ?????? h?0
?
1 + 1
h
= 0 ?? '
(1) = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
2{1 + h}  1 + 1
h
= ?????? h?0
?
2h
h
= 2
Hence ?? (?? ) is not differentiable at ?? = 1.
4. IMPORTANT NOTE:
(a) Let ????
'
(?? ) = ?? &????
'
(?? ) = ?? where ?? &?? are finite then:
(i) ?? = ?? ? ?? is differentiable at ?? = ?? ? ?? is continuous at ?? = ??
(ii) ?? ? ?? ? ?? is not differentiable at ?? = ?? , but ?? is continuous at ?? = ?? .
Problem 4: Determine the values of ?? for which the following functions fails to be continuous or
differentiable ?? (?? ) = {(1  ?? ), ?? < 1 (1  ?? )(2  ?? ), 1 = ?? = 2 (3  ?? ), ?? > 2 , Justify your answer.
Solution: By the given definition it is clear that the function ?? is continuous and differentiable at all
points except possibily at ?? = 1 and ?? = 2.
Check the differentiability at ?? = 1
?? = ?????? = ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
1  (1  h)  0
h
= 1 ?? = ?????? = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
{1  (1 + h)}{2  (1 + h)}  0
h
= 1 ? ?? = ?? ? ???????????????????????????? ???? ?? = 1.
? ???????????????????? ???? ?? = 1.
Check the differentiability at ?? = 2
?? = ?????? = ?????? h?0
?
?? (2  h)  ?? (2)
h
= ?????? h?0
?
(1  2 + h)(2  2 + h)  0
h
= 1 = ???????????? ?? = ?????? = ?????? h?0
?
?? (2 + h)  ?? (2)
h
= ?????? h?0
?
(3  2  h)  0
h
? 8 (?????? ???????????? ) ? ?? ? ??
? ?????? ???????????????????????????? ???? ?? = 2.
Now we have to check the continuity at ?? = 2
?????? = ?????? ?? ?2

??? (?? ) = ?????? ?? ?2

?(1  ?? )(2  ?? ) = ?????? h?0
?(1  (2  h))(2  (2  h)) = 0 ?????? = ?????? ?? ?2
+
??? (?? )
= ?????? ?? ?2
+
?(3  ?? ) = ?????? h?0
?(3  (2 + h)) = 1 ? ?????? ? ?????? ? ?????? ???????????????????? ???? ?? = 2.
(b) Vertical tangent:
(i) If ?? = ?? (?? ) is continuous at ?? = ?? and ?????? ?? ??? ??? '
(?? ) approaches to 8, then ?? = ?? (?? ) has a vertical
tangent at ?? = ?? . If a function has vertical tangent at ?? = ?? then it is non differentiable at ?? = ?? .
e.g. (1) ?? (?? ) = ?? 1/3
has vertical tangent at ?? = 0
Page 5
DIFFERENTIABILITY
Differentiability is a fundamental concept in calculus that extends the idea of continuity. It focuses on
how smoothly a function changes at a given point, providing insight into the function's rate of change
and local behavior. This concept is crucial for analyzing functions, solving optimization problems, and
applying calculus across various fields of science and engineering.
This section delves into the meaning of derivatives, explores the criteria for differentiability, and
examines its relationship with continuity. Through a combination of theoretical explanations and
practical examples, we'll investigate the nuances of differentiability, including special cases and
geometric interpretations. By understanding differentiability, we gain powerful tools for mathematical
analysis and problemsolving in calculus.
1. MEANING OF DERIVATIVE:
The instantaneous rate of change of a function with respect to the dependent variable is called derivative.
Let ' ?? ' be a given function of one variable and let ?? ?? denote a number (positive or negative) to be added
to the number ?? . Let ?? ?? denote the corresponding change of ' ?? ' then ?? ?? = ?? (?? + ???? )  ?? (?? )
?
????
????
=
?? (?? + ???? )  ?? (?? )
????
If ???? /???? approaches a limit as ???? approaches zero, this limit is the derivative of ' ?? ' at the point ?? . The
derivative of a function ' ?? ' is a function ; this function is denoted by symbols such as
?? '
(?? ),
????
????
,
?? ????
?? (?? ) ????
???? (?? )
????
?
????
????
= ?????? ???? ?0
?
????
????
= ?????? ???? ?0
?
?? (?? + ???? )  ?? (?? )
????
The derivative evaluated at a point ?? , is written, ?? '
(?? ),
???? (?? )
????

?? =?? , ?? '
(?? )
?? =?? , etc.
2. EXISTENCE OF DERIVATIVE AT x= a
(a) Right hand derivative:
The right hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
?? ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
, provided the limit exists & is finite. ( h > 0 )
(b) Left hand derivative:
The left hand derivative of ?? (?? ) at ?? = a denoted by ????
'
(?? ) is defined as:
????
'
(?? ) =
h?0
?? (?? h)?? (?? )
h
, provided the limit exists & is finite. (h > 0 )
Hence ?? (?? ) is said to be derivable or differentiable at ?? = ?? . If ????
'
(?? ) = ????
'
(?? ) = finite quantity
and it is denoted by ?? (?? ); where ?? (?? ) = ????
'
(?? ) = ????
'
(a) & it is called derivative or differential
coefficient of ?? (?? ) at ?? = ?? .
3. DIFFERENTIABILITY & CONTINUITY:
Theorem: If a function ?? (?? ) is derivable at ?? = ?? , then ?? (?? ) is continuous at ?? = ?? .
Proof: ?? '
(?? ) = ?????? h?0
?? (?? +h)?? (?? )
h
exists.
Also ?? (?? + h)  ?? (?? ) =
?? (?? +h)?? (?? )
h
· h [h ? 0]
? ?????? h?0
[?? (?? + h)  ?? (?? )] = ?????? h?0
?? (?? + h)  ?? (?? )
h
· h = ?? '
(?? ) · 0 = 0
? ?????? h?0
[?? (?? + h)  ?? (?? )] = 0 ? ?????? h?0
?? (?? + h) = ?? (?? ) ? ?? (?? ) is continuous at ?? = ?? .
Note:
(i) ???????????????????????????? ? ???????????????????? ; ???????????????????? ?? ???????????????????????????? ; ?????? ???????????????????????????? ?
? ?????? ???????????????????? ?????? ?????? ???????????????????? ? ?????? ????????????????????????????
(ii) All polynomial, rational, trigonometric, logarithmic and exponential function are continuous and
differentiable in their domains.
(iii) If ?? (?? )&?? (?? ) are differentiable at ?? = a, then the functions ?? (?? ) + ?? (?? ), ?? (?? )  ?? (?? ), ?? (?? ). ?? (?? ) will
also be differentiable at ?? = ?? & if ?? (?? ) ? 0 then the function ?? (?? )/?? (?? ) will also be differentiable at ?? =
?? .
?? = 0&
?? 2
Solution:
?? (?? ) = {1 + ?? ; 8 < ?? < 0  1 + ?????? ?? ; 0 = ?? <
?? 2
?????? ?? ;
?? 2
= ?? < 8
To check the differentiability at ?? = 0
?????? = ?????? h?0
?
?? (0  h)  ?? (0)
h
= ?????? h?0
?
1 + 0  h  (1)
h
= 1
?????? = ?????? h?0
?
?? (0 + h)  ?? (0)
h
= ?????? h?0
?
1 + ?????? h + 1
h
= 1 ? ?????? = ?????? ? ???????????????????????????? ???? ?? = 0.
? ???????????????????? ???? ?? = 0.
To check the continuity at ?? =
?? 2
LHL ?????? ?? ?
?? 2

??? (?? ) = ?????? ?? ?
?? 
2
?(1 + ?????? ?? ) = 0
RHL ?????? ?? ?
?? +
2
??? (?? ) = ?????? ?? ?
?? +
2
??????? ?? = 0
? LHL = RHL = ?? (
?? 2
) = 0
? Continuous at ?? =
?? 2
.
To check the differentiability at ?? =
?? 2
?????? = ?????? h?0
?
?? (
?? 2
 h)  ?? (
?? 2
)
h
= ?????? h?0
?
1 + ?????? h  0
h
= 0 ?????? = ?????? h?0
?
?? (
?? 2
+ h)  ?? (
?? 2
)
h
= ?????? h?0
?
?????? h  0
h
= 1 ? ?????? ? ?????? ? ?????? ???????????????????????????? ???? ?? =
?? 2
.
Problem 2: If ?? (?? ) = {?? + ?? ?? 2
?? < 1 3????  ?? + 2?? = 1
then find ?? and ?? so that ?? (?? ) become differentiable at ?? = 1.
Solution:
????
'
(1) = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
3?? (1 + h)  ?? + 2  3?? + ??  2
h
= ?????? h?0
?
3?? h
h
= 3?? ?? ?? '
(1)
= ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
?? + ?? (1  h)
2
 3?? + ??  2
h
= ?????? h?0
?
(2?? + 2??  2) + ?? h
2
 2?? h
h
hence for this limit to be defined
2?? + 2??  2 = 0 ?? = ?? + 1 ?? ?? '
(1) = ?????? h?0
?  (?? h  2?? ) = 2?? ? ?? ?? '
(1) = ?? ?? '
(1) 3?? = 2?? = 2(?? + 1) ?? = 2, ?? = 3
Problem 3: ?? (?? ) = {[?????? ???? ] ?? = 1 2{?? }  1 ?? > 1 comment on the derivability at ?? = 1, where [ ]
denotes greatest integer function &{ }enotes fractional part function.
Solution:
?? ?? '
(1) = ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
[?????? (??  ?? h)] + 1
h
= ?????? h?0
?
1 + 1
h
= 0 ?? '
(1) = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
2{1 + h}  1 + 1
h
= ?????? h?0
?
2h
h
= 2
Hence ?? (?? ) is not differentiable at ?? = 1.
4. IMPORTANT NOTE:
(a) Let ????
'
(?? ) = ?? &????
'
(?? ) = ?? where ?? &?? are finite then:
(i) ?? = ?? ? ?? is differentiable at ?? = ?? ? ?? is continuous at ?? = ??
(ii) ?? ? ?? ? ?? is not differentiable at ?? = ?? , but ?? is continuous at ?? = ?? .
Problem 4: Determine the values of ?? for which the following functions fails to be continuous or
differentiable ?? (?? ) = {(1  ?? ), ?? < 1 (1  ?? )(2  ?? ), 1 = ?? = 2 (3  ?? ), ?? > 2 , Justify your answer.
Solution: By the given definition it is clear that the function ?? is continuous and differentiable at all
points except possibily at ?? = 1 and ?? = 2.
Check the differentiability at ?? = 1
?? = ?????? = ?????? h?0
?
?? (1  h)  ?? (1)
h
= ?????? h?0
?
1  (1  h)  0
h
= 1 ?? = ?????? = ?????? h?0
?
?? (1 + h)  ?? (1)
h
= ?????? h?0
?
{1  (1 + h)}{2  (1 + h)}  0
h
= 1 ? ?? = ?? ? ???????????????????????????? ???? ?? = 1.
? ???????????????????? ???? ?? = 1.
Check the differentiability at ?? = 2
?? = ?????? = ?????? h?0
?
?? (2  h)  ?? (2)
h
= ?????? h?0
?
(1  2 + h)(2  2 + h)  0
h
= 1 = ???????????? ?? = ?????? = ?????? h?0
?
?? (2 + h)  ?? (2)
h
= ?????? h?0
?
(3  2  h)  0
h
? 8 (?????? ???????????? ) ? ?? ? ??
? ?????? ???????????????????????????? ???? ?? = 2.
Now we have to check the continuity at ?? = 2
?????? = ?????? ?? ?2

??? (?? ) = ?????? ?? ?2

?(1  ?? )(2  ?? ) = ?????? h?0
?(1  (2  h))(2  (2  h)) = 0 ?????? = ?????? ?? ?2
+
??? (?? )
= ?????? ?? ?2
+
?(3  ?? ) = ?????? h?0
?(3  (2 + h)) = 1 ? ?????? ? ?????? ? ?????? ???????????????????? ???? ?? = 2.
(b) Vertical tangent:
(i) If ?? = ?? (?? ) is continuous at ?? = ?? and ?????? ?? ??? ??? '
(?? ) approaches to 8, then ?? = ?? (?? ) has a vertical
tangent at ?? = ?? . If a function has vertical tangent at ?? = ?? then it is non differentiable at ?? = ?? .
e.g. (1) ?? (?? ) = ?? 1/3
has vertical tangent at ?? = 0
since ?? +
'
(0) ? 8 and ?? 
'
(0) ? 8 hence ?? (?? ) is not differentiable at ?? = 0
(2) ?? (?? ) = ?? 2/3
have vertical tangent at ?? = 0 since ?? +
'
(0) ? 8 and ?? 
'
(0) ? 8 hence ?? (?? ) is not
differentiable at ?? = 0.
(c) Geometrical interpretation of differentiability:
(i) If the function ?? = ?? (?? ) is differentiable at ?? = ?? , then a unique non vertical tangent can be drawn to
the curve ?? = ?? (?? ) at the point ?? (?? , ?? (?? ))&?? '
(?? ) represent the slope of the tangent at point ?? .
(ii) If a function ?? (?? ) does not have a unique tangent ( ?????? are finite but unequal), then ?? is continuous
at ?? = ?? , it geometrically implies a corner at ?? = ?? .
e.g. ?? (?? ) = ??  is continuous but not differentiable at ?? = 0 & there is corner at ?? = 0.
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