Detailed Notes: Differentiability | Mathematics (Maths) for JEE Main & Advanced PDF Download

 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 problem-solving 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 problem-solving 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 problem-solving 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 problem-solving 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 problem-solving 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.