Detailed Notes: Applications of Derivatives | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Introduction 
Imagine you're on a road trip, and you’re curious about how fast you're traveling at any 
given moment. You look at your speedometer, which tells you your speed right now, not 
just your average speed over the whole trip. This immediate speed is what we call the 
instantaneous rate of change in mathematics, and it’s found using a powerful tool called 
differentiation. 
The Concept of Change 
In everyday life, we're constantly observing changes: the change in weather, the change 
in stock prices, or even the change in how quickly you finish a game as you get better. 
Differentiation is the mathematical way of understanding how things change.  
The Slope of a Curve 
To understand differentiation, think of a curve on a graph. The slope of the curve at any 
point tells you how steep the curve is at that exact spot. If you were skiing down a 
mountain, the slope would tell you how fast you'd be going and how much you'd have to 
lean to stay upright. In math terms, this slope is called the derivative. 
METHODS OF DIFFERENTIATION 
1. Derivative from the First Principle 
One of the fundamental ways to find this slope (or derivative) is using the first 
principle of differentiation. Here’s a simple way to grasp it: 
Page 2


Introduction 
Imagine you're on a road trip, and you’re curious about how fast you're traveling at any 
given moment. You look at your speedometer, which tells you your speed right now, not 
just your average speed over the whole trip. This immediate speed is what we call the 
instantaneous rate of change in mathematics, and it’s found using a powerful tool called 
differentiation. 
The Concept of Change 
In everyday life, we're constantly observing changes: the change in weather, the change 
in stock prices, or even the change in how quickly you finish a game as you get better. 
Differentiation is the mathematical way of understanding how things change.  
The Slope of a Curve 
To understand differentiation, think of a curve on a graph. The slope of the curve at any 
point tells you how steep the curve is at that exact spot. If you were skiing down a 
mountain, the slope would tell you how fast you'd be going and how much you'd have to 
lean to stay upright. In math terms, this slope is called the derivative. 
METHODS OF DIFFERENTIATION 
1. Derivative from the First Principle 
One of the fundamental ways to find this slope (or derivative) is using the first 
principle of differentiation. Here’s a simple way to grasp it: 
1. Function and Points: Imagine you have a function y = f(x), which could be 
anything from a simple line to a complex curve. Pick a point on this function, say 
x. 
2. Tiny Change: Consider a tiny change in x, let’s call it ???? . Now look at how the 
function changes with this tiny change in x. 
3. Slope Calculation: The slope of the function at x is then given by the formula: 
?????? ???? ?0
????
????
= ?????? ???? ?0
?? ( ?? +???? ) -?? ( ?? )
????
= ?? '
( ?? )=
????
????
  
Note : 
????
????
 can also be represented as ?? 1
 or ?? '
 or Dy or ?? '
( ?? ) . 
????
????
 represents instantaneous 
rate of change of ?? w.r.t. ?? . 
Problem 1 :   Differentiate each of following functions by first principle : 
 ( ?? ) ?? ( ?? )= ?????? ??  ( ???? ) ?? ( ?? )= ?? ?????? ??  
Solution : 
(i)  ?? '
( ?? )= ?????? h?0
?
?????? ( ?? +h) -?????? ?? h
= ?????? h?0
?
?????? ( ?? +h-?? ) [1+?????? ?? ?????? ( ?? +h) ]
h
 
= ?????? h?0
?
?????? h
h
· ( 1 + ?????? 2
 ?? )= ?????? 2
 ?? . 
(ii) 
?? '
( ?? )  = ?????? h?0
?
?? ?????? ( ?? +h)
- ?? ?????? ?? h
= ?????? h?0
??? ?????? ?? [?? ?????? ( ?? +h) -?????? ?? - 1]
?????? ( ?? + h)- ?????? ?? (
?????? ( ?? + h)- ?????? ?? h
)   
= ?? ?????? ?? ?????? h?0
?
?????? ( ?? + h)- ?????? ?? h
= ?? ?????? ?? ?????? ??  
2. DERIVATIVE OF STANDARD FUNCTIONS : 
 ?? ( ?? ) ?? '
( ?? )  ?? ( ?? ) ?? '
( ?? ) 
(i) ?? ?? ?? ?? ?? -1
 (ii) ?? ?? ?? ?? 
(iii) ?? ?? ?? ?? ?????? , ?? > 0 (iv) ?????? 1/?? 
(v) ??????
?? ?? 
( 1/?? ) ??????
?? ?? , ?? > 0, ?? ? 1 
(vi) ?????? ?? ?????? ?? 
(vii) ?????? ?? -?????? ?? (viii) ?????? ?? ?????? 2
 ?? 
(ix) ?????? ?? ?????? ???????? ?? (x) cosecx - cosecx . cotx 
Page 3


Introduction 
Imagine you're on a road trip, and you’re curious about how fast you're traveling at any 
given moment. You look at your speedometer, which tells you your speed right now, not 
just your average speed over the whole trip. This immediate speed is what we call the 
instantaneous rate of change in mathematics, and it’s found using a powerful tool called 
differentiation. 
The Concept of Change 
In everyday life, we're constantly observing changes: the change in weather, the change 
in stock prices, or even the change in how quickly you finish a game as you get better. 
Differentiation is the mathematical way of understanding how things change.  
The Slope of a Curve 
To understand differentiation, think of a curve on a graph. The slope of the curve at any 
point tells you how steep the curve is at that exact spot. If you were skiing down a 
mountain, the slope would tell you how fast you'd be going and how much you'd have to 
lean to stay upright. In math terms, this slope is called the derivative. 
METHODS OF DIFFERENTIATION 
1. Derivative from the First Principle 
One of the fundamental ways to find this slope (or derivative) is using the first 
principle of differentiation. Here’s a simple way to grasp it: 
1. Function and Points: Imagine you have a function y = f(x), which could be 
anything from a simple line to a complex curve. Pick a point on this function, say 
x. 
2. Tiny Change: Consider a tiny change in x, let’s call it ???? . Now look at how the 
function changes with this tiny change in x. 
3. Slope Calculation: The slope of the function at x is then given by the formula: 
?????? ???? ?0
????
????
= ?????? ???? ?0
?? ( ?? +???? ) -?? ( ?? )
????
= ?? '
( ?? )=
????
????
  
Note : 
????
????
 can also be represented as ?? 1
 or ?? '
 or Dy or ?? '
( ?? ) . 
????
????
 represents instantaneous 
rate of change of ?? w.r.t. ?? . 
Problem 1 :   Differentiate each of following functions by first principle : 
 ( ?? ) ?? ( ?? )= ?????? ??  ( ???? ) ?? ( ?? )= ?? ?????? ??  
Solution : 
(i)  ?? '
( ?? )= ?????? h?0
?
?????? ( ?? +h) -?????? ?? h
= ?????? h?0
?
?????? ( ?? +h-?? ) [1+?????? ?? ?????? ( ?? +h) ]
h
 
= ?????? h?0
?
?????? h
h
· ( 1 + ?????? 2
 ?? )= ?????? 2
 ?? . 
(ii) 
?? '
( ?? )  = ?????? h?0
?
?? ?????? ( ?? +h)
- ?? ?????? ?? h
= ?????? h?0
??? ?????? ?? [?? ?????? ( ?? +h) -?????? ?? - 1]
?????? ( ?? + h)- ?????? ?? (
?????? ( ?? + h)- ?????? ?? h
)   
= ?? ?????? ?? ?????? h?0
?
?????? ( ?? + h)- ?????? ?? h
= ?? ?????? ?? ?????? ??  
2. DERIVATIVE OF STANDARD FUNCTIONS : 
 ?? ( ?? ) ?? '
( ?? )  ?? ( ?? ) ?? '
( ?? ) 
(i) ?? ?? ?? ?? ?? -1
 (ii) ?? ?? ?? ?? 
(iii) ?? ?? ?? ?? ?????? , ?? > 0 (iv) ?????? 1/?? 
(v) ??????
?? ?? 
( 1/?? ) ??????
?? ?? , ?? > 0, ?? ? 1 
(vi) ?????? ?? ?????? ?? 
(vii) ?????? ?? -?????? ?? (viii) ?????? ?? ?????? 2
 ?? 
(ix) ?????? ?? ?????? ???????? ?? (x) cosecx - cosecx . cotx 
(xi) ???????? -?????????? 2
 ?? (xii) constant  
(xiii) ?????? -1
 ?? 
1
v 1 - ?? 2
, -1 < ?? < 1 
(xiv) ?????? -1
 ?? 
-1
v 1 - ?? 2
, -1 < ?? < 1 
(xv) ?????? -1
 ?? 
1
1 + ?? 2
, ?? ? ?? (xvi) ?????? -1
 ?? 
1
|?? |v ?? 2
- 1
, |?? |
> 1 
(xvii) ?????????? -1
 ?? 
-1
|?? |v ?? 2
- 1
, |?? | > 1 
(xviii) ?????? -1
 ?? 
-1
1 + ?? 2
, ?? ? ?? 
 
3. FUNDAMENTAL THEOREMS : 
Just like in arithmetic, we have fundamental operations, in differentiation, we have 
fundamental theorems. If ?? and ?? are derivable functions of ?? , then, 
(a) 
?? ????
( ?? ± ?? )=
????
????
±
????
????
 
(b) 
?? ????
( ???? )= ?? ????
????
, where ?? is any constant 
(c) 
?? ????
( ???? )= ?? ????
????
+ ?? ????
????
 known as "PRODUCT RULE" 
(d) 
?? ????
(
?? ?? )=
?? (
????
????
) -?? (
????
????
)
?? 2
 where ?? ? 0 known as "QUOTIENT RULE" 
(e) If ?? = ?? ( ?? ) & ?? = ?? ( ?? ) then 
????
????
=
????
????
·
????
????
 known as "CHAIN RULE" 
Note : In general if ?? = ?? ( ?? ) then 
????
????
= ?? '
( ?? )·
????
????
. 
Problem 2: If ?? = ?? ?? ?????? ?? + ???????? ?? ?? , find 
????
????
. 
Solution:  ?? = ?? ?? .tan ?? + ?? · ??????
?? ?? 
On differentiating we get, 
????
????
= ?? ?? · ?????? ?? + ?? ?? · ?????? 2
 ?? + 1 · ?????? ?? + ?? ·
1
?? 
Hence, 
????
????
= ?? ?? ( ?????? ?? + ?????? 2
 ?? )+ ( ?????? ?? + 1) 
Problem 3 : If ?? =
?????? ?? ?? + ?? ?? ?????? 2?? + ?????? 5
 ?? , find 
????
????
. 
Solution : On differentiating we get, 
Page 4


Introduction 
Imagine you're on a road trip, and you’re curious about how fast you're traveling at any 
given moment. You look at your speedometer, which tells you your speed right now, not 
just your average speed over the whole trip. This immediate speed is what we call the 
instantaneous rate of change in mathematics, and it’s found using a powerful tool called 
differentiation. 
The Concept of Change 
In everyday life, we're constantly observing changes: the change in weather, the change 
in stock prices, or even the change in how quickly you finish a game as you get better. 
Differentiation is the mathematical way of understanding how things change.  
The Slope of a Curve 
To understand differentiation, think of a curve on a graph. The slope of the curve at any 
point tells you how steep the curve is at that exact spot. If you were skiing down a 
mountain, the slope would tell you how fast you'd be going and how much you'd have to 
lean to stay upright. In math terms, this slope is called the derivative. 
METHODS OF DIFFERENTIATION 
1. Derivative from the First Principle 
One of the fundamental ways to find this slope (or derivative) is using the first 
principle of differentiation. Here’s a simple way to grasp it: 
1. Function and Points: Imagine you have a function y = f(x), which could be 
anything from a simple line to a complex curve. Pick a point on this function, say 
x. 
2. Tiny Change: Consider a tiny change in x, let’s call it ???? . Now look at how the 
function changes with this tiny change in x. 
3. Slope Calculation: The slope of the function at x is then given by the formula: 
?????? ???? ?0
????
????
= ?????? ???? ?0
?? ( ?? +???? ) -?? ( ?? )
????
= ?? '
( ?? )=
????
????
  
Note : 
????
????
 can also be represented as ?? 1
 or ?? '
 or Dy or ?? '
( ?? ) . 
????
????
 represents instantaneous 
rate of change of ?? w.r.t. ?? . 
Problem 1 :   Differentiate each of following functions by first principle : 
 ( ?? ) ?? ( ?? )= ?????? ??  ( ???? ) ?? ( ?? )= ?? ?????? ??  
Solution : 
(i)  ?? '
( ?? )= ?????? h?0
?
?????? ( ?? +h) -?????? ?? h
= ?????? h?0
?
?????? ( ?? +h-?? ) [1+?????? ?? ?????? ( ?? +h) ]
h
 
= ?????? h?0
?
?????? h
h
· ( 1 + ?????? 2
 ?? )= ?????? 2
 ?? . 
(ii) 
?? '
( ?? )  = ?????? h?0
?
?? ?????? ( ?? +h)
- ?? ?????? ?? h
= ?????? h?0
??? ?????? ?? [?? ?????? ( ?? +h) -?????? ?? - 1]
?????? ( ?? + h)- ?????? ?? (
?????? ( ?? + h)- ?????? ?? h
)   
= ?? ?????? ?? ?????? h?0
?
?????? ( ?? + h)- ?????? ?? h
= ?? ?????? ?? ?????? ??  
2. DERIVATIVE OF STANDARD FUNCTIONS : 
 ?? ( ?? ) ?? '
( ?? )  ?? ( ?? ) ?? '
( ?? ) 
(i) ?? ?? ?? ?? ?? -1
 (ii) ?? ?? ?? ?? 
(iii) ?? ?? ?? ?? ?????? , ?? > 0 (iv) ?????? 1/?? 
(v) ??????
?? ?? 
( 1/?? ) ??????
?? ?? , ?? > 0, ?? ? 1 
(vi) ?????? ?? ?????? ?? 
(vii) ?????? ?? -?????? ?? (viii) ?????? ?? ?????? 2
 ?? 
(ix) ?????? ?? ?????? ???????? ?? (x) cosecx - cosecx . cotx 
(xi) ???????? -?????????? 2
 ?? (xii) constant  
(xiii) ?????? -1
 ?? 
1
v 1 - ?? 2
, -1 < ?? < 1 
(xiv) ?????? -1
 ?? 
-1
v 1 - ?? 2
, -1 < ?? < 1 
(xv) ?????? -1
 ?? 
1
1 + ?? 2
, ?? ? ?? (xvi) ?????? -1
 ?? 
1
|?? |v ?? 2
- 1
, |?? |
> 1 
(xvii) ?????????? -1
 ?? 
-1
|?? |v ?? 2
- 1
, |?? | > 1 
(xviii) ?????? -1
 ?? 
-1
1 + ?? 2
, ?? ? ?? 
 
3. FUNDAMENTAL THEOREMS : 
Just like in arithmetic, we have fundamental operations, in differentiation, we have 
fundamental theorems. If ?? and ?? are derivable functions of ?? , then, 
(a) 
?? ????
( ?? ± ?? )=
????
????
±
????
????
 
(b) 
?? ????
( ???? )= ?? ????
????
, where ?? is any constant 
(c) 
?? ????
( ???? )= ?? ????
????
+ ?? ????
????
 known as "PRODUCT RULE" 
(d) 
?? ????
(
?? ?? )=
?? (
????
????
) -?? (
????
????
)
?? 2
 where ?? ? 0 known as "QUOTIENT RULE" 
(e) If ?? = ?? ( ?? ) & ?? = ?? ( ?? ) then 
????
????
=
????
????
·
????
????
 known as "CHAIN RULE" 
Note : In general if ?? = ?? ( ?? ) then 
????
????
= ?? '
( ?? )·
????
????
. 
Problem 2: If ?? = ?? ?? ?????? ?? + ???????? ?? ?? , find 
????
????
. 
Solution:  ?? = ?? ?? .tan ?? + ?? · ??????
?? ?? 
On differentiating we get, 
????
????
= ?? ?? · ?????? ?? + ?? ?? · ?????? 2
 ?? + 1 · ?????? ?? + ?? ·
1
?? 
Hence, 
????
????
= ?? ?? ( ?????? ?? + ?????? 2
 ?? )+ ( ?????? ?? + 1) 
Problem 3 : If ?? =
?????? ?? ?? + ?? ?? ?????? 2?? + ?????? 5
 ?? , find 
????
????
. 
Solution : On differentiating we get, 
????
????
=
?? ????
(
?????? ?? ?? )+
?? ????
( ?? ?? ?????? 2?? )+
?? ????
( ??????
5
 ?? )
=
1
?? · ?? - ?????? ?? · 1
?? 2
+ ?? ?? ?????? 2?? + 2?? ?? · ?????? 2?? +
1
?? ??????
?? 5
 
Hence,  
????
????
= (
1-?????? ?? ?? 2
)+ ?? ?? ( ?????? 2?? + 2?????? 2?? )+
1
?? ?????? ?? 5
 
Problem 4 : If ?? = ??????
?? ( ?????? -1
 v 1 + ?? 2
) , find 
????
????
. 
Solution :  ?? = ?????? ?? ( ?????? -1
 v 1 + ?? 2
) 
On differentiating we get, 
  =
1
?????? -1
 v 1 + ?? 2
·
1
1 + ( v 1 + ?? 2
)
2
·
1
2v 1 + ?? 2
· 2??   
=
?? ( ?????? -1
 v 1 + ?? 2
){1 + ( v 1 + ?? 2
)
2
}v 1 + ?? 2
=
?? ( ?????? -1
 v 1 + ?? 2
) ( 2 + ?? 2
) v 1 + ?? 2
  
4. LOGARITHMIC DIFFERENTIATION : 
To find the derivative of a function : 
(a) which is the product or quotient of a number of functions or 
(b) of the form [?? ( ?? ) ]
?? ( ?? )
 where ?? & ?? are both derivable functions. 
It is convenient to take the logarithm of the function first & then differentiate. 
Problem 5: If ?? = ( ?????? ?? )
???? ?? , find 
????
????
 
Solution :  ?????? = ?????? . ???? ( ?????? ?? ) 
On differentiating we get, 
1
?? ????
????
=
1
?? ???? ( ?????? ?? )+ ?????? .
?????? ?? ?????? ?? ?
????
????
= ( ?????? ?? )
???? ?? [
?????? ( ?????? ?? )
?? + ?????? ?????? ?? ]  Ans. 
Problem 6 : If ?? = ?????? (?????? -1
 (
?? -?? 2
?? 2
) ), then 
????
????
 equals - 
(A) ?? [1 + ?????? ( ?????? ?? )+ ?????? 2
 ?? ] 
(B) 2?? [1 + ?????? ( ?????? ?? ) ] + ?????? 2
 ?? 
(C) 2?? [1 + ?????? ( ?????? ?? ) ] + ?????? ?? 
(D) 2?? + ?? [1 + ?????? ( ?????? ?? ) ]
2
 
Solution : Taking log on both sides, we get 
Page 5


Introduction 
Imagine you're on a road trip, and you’re curious about how fast you're traveling at any 
given moment. You look at your speedometer, which tells you your speed right now, not 
just your average speed over the whole trip. This immediate speed is what we call the 
instantaneous rate of change in mathematics, and it’s found using a powerful tool called 
differentiation. 
The Concept of Change 
In everyday life, we're constantly observing changes: the change in weather, the change 
in stock prices, or even the change in how quickly you finish a game as you get better. 
Differentiation is the mathematical way of understanding how things change.  
The Slope of a Curve 
To understand differentiation, think of a curve on a graph. The slope of the curve at any 
point tells you how steep the curve is at that exact spot. If you were skiing down a 
mountain, the slope would tell you how fast you'd be going and how much you'd have to 
lean to stay upright. In math terms, this slope is called the derivative. 
METHODS OF DIFFERENTIATION 
1. Derivative from the First Principle 
One of the fundamental ways to find this slope (or derivative) is using the first 
principle of differentiation. Here’s a simple way to grasp it: 
1. Function and Points: Imagine you have a function y = f(x), which could be 
anything from a simple line to a complex curve. Pick a point on this function, say 
x. 
2. Tiny Change: Consider a tiny change in x, let’s call it ???? . Now look at how the 
function changes with this tiny change in x. 
3. Slope Calculation: The slope of the function at x is then given by the formula: 
?????? ???? ?0
????
????
= ?????? ???? ?0
?? ( ?? +???? ) -?? ( ?? )
????
= ?? '
( ?? )=
????
????
  
Note : 
????
????
 can also be represented as ?? 1
 or ?? '
 or Dy or ?? '
( ?? ) . 
????
????
 represents instantaneous 
rate of change of ?? w.r.t. ?? . 
Problem 1 :   Differentiate each of following functions by first principle : 
 ( ?? ) ?? ( ?? )= ?????? ??  ( ???? ) ?? ( ?? )= ?? ?????? ??  
Solution : 
(i)  ?? '
( ?? )= ?????? h?0
?
?????? ( ?? +h) -?????? ?? h
= ?????? h?0
?
?????? ( ?? +h-?? ) [1+?????? ?? ?????? ( ?? +h) ]
h
 
= ?????? h?0
?
?????? h
h
· ( 1 + ?????? 2
 ?? )= ?????? 2
 ?? . 
(ii) 
?? '
( ?? )  = ?????? h?0
?
?? ?????? ( ?? +h)
- ?? ?????? ?? h
= ?????? h?0
??? ?????? ?? [?? ?????? ( ?? +h) -?????? ?? - 1]
?????? ( ?? + h)- ?????? ?? (
?????? ( ?? + h)- ?????? ?? h
)   
= ?? ?????? ?? ?????? h?0
?
?????? ( ?? + h)- ?????? ?? h
= ?? ?????? ?? ?????? ??  
2. DERIVATIVE OF STANDARD FUNCTIONS : 
 ?? ( ?? ) ?? '
( ?? )  ?? ( ?? ) ?? '
( ?? ) 
(i) ?? ?? ?? ?? ?? -1
 (ii) ?? ?? ?? ?? 
(iii) ?? ?? ?? ?? ?????? , ?? > 0 (iv) ?????? 1/?? 
(v) ??????
?? ?? 
( 1/?? ) ??????
?? ?? , ?? > 0, ?? ? 1 
(vi) ?????? ?? ?????? ?? 
(vii) ?????? ?? -?????? ?? (viii) ?????? ?? ?????? 2
 ?? 
(ix) ?????? ?? ?????? ???????? ?? (x) cosecx - cosecx . cotx 
(xi) ???????? -?????????? 2
 ?? (xii) constant  
(xiii) ?????? -1
 ?? 
1
v 1 - ?? 2
, -1 < ?? < 1 
(xiv) ?????? -1
 ?? 
-1
v 1 - ?? 2
, -1 < ?? < 1 
(xv) ?????? -1
 ?? 
1
1 + ?? 2
, ?? ? ?? (xvi) ?????? -1
 ?? 
1
|?? |v ?? 2
- 1
, |?? |
> 1 
(xvii) ?????????? -1
 ?? 
-1
|?? |v ?? 2
- 1
, |?? | > 1 
(xviii) ?????? -1
 ?? 
-1
1 + ?? 2
, ?? ? ?? 
 
3. FUNDAMENTAL THEOREMS : 
Just like in arithmetic, we have fundamental operations, in differentiation, we have 
fundamental theorems. If ?? and ?? are derivable functions of ?? , then, 
(a) 
?? ????
( ?? ± ?? )=
????
????
±
????
????
 
(b) 
?? ????
( ???? )= ?? ????
????
, where ?? is any constant 
(c) 
?? ????
( ???? )= ?? ????
????
+ ?? ????
????
 known as "PRODUCT RULE" 
(d) 
?? ????
(
?? ?? )=
?? (
????
????
) -?? (
????
????
)
?? 2
 where ?? ? 0 known as "QUOTIENT RULE" 
(e) If ?? = ?? ( ?? ) & ?? = ?? ( ?? ) then 
????
????
=
????
????
·
????
????
 known as "CHAIN RULE" 
Note : In general if ?? = ?? ( ?? ) then 
????
????
= ?? '
( ?? )·
????
????
. 
Problem 2: If ?? = ?? ?? ?????? ?? + ???????? ?? ?? , find 
????
????
. 
Solution:  ?? = ?? ?? .tan ?? + ?? · ??????
?? ?? 
On differentiating we get, 
????
????
= ?? ?? · ?????? ?? + ?? ?? · ?????? 2
 ?? + 1 · ?????? ?? + ?? ·
1
?? 
Hence, 
????
????
= ?? ?? ( ?????? ?? + ?????? 2
 ?? )+ ( ?????? ?? + 1) 
Problem 3 : If ?? =
?????? ?? ?? + ?? ?? ?????? 2?? + ?????? 5
 ?? , find 
????
????
. 
Solution : On differentiating we get, 
????
????
=
?? ????
(
?????? ?? ?? )+
?? ????
( ?? ?? ?????? 2?? )+
?? ????
( ??????
5
 ?? )
=
1
?? · ?? - ?????? ?? · 1
?? 2
+ ?? ?? ?????? 2?? + 2?? ?? · ?????? 2?? +
1
?? ??????
?? 5
 
Hence,  
????
????
= (
1-?????? ?? ?? 2
)+ ?? ?? ( ?????? 2?? + 2?????? 2?? )+
1
?? ?????? ?? 5
 
Problem 4 : If ?? = ??????
?? ( ?????? -1
 v 1 + ?? 2
) , find 
????
????
. 
Solution :  ?? = ?????? ?? ( ?????? -1
 v 1 + ?? 2
) 
On differentiating we get, 
  =
1
?????? -1
 v 1 + ?? 2
·
1
1 + ( v 1 + ?? 2
)
2
·
1
2v 1 + ?? 2
· 2??   
=
?? ( ?????? -1
 v 1 + ?? 2
){1 + ( v 1 + ?? 2
)
2
}v 1 + ?? 2
=
?? ( ?????? -1
 v 1 + ?? 2
) ( 2 + ?? 2
) v 1 + ?? 2
  
4. LOGARITHMIC DIFFERENTIATION : 
To find the derivative of a function : 
(a) which is the product or quotient of a number of functions or 
(b) of the form [?? ( ?? ) ]
?? ( ?? )
 where ?? & ?? are both derivable functions. 
It is convenient to take the logarithm of the function first & then differentiate. 
Problem 5: If ?? = ( ?????? ?? )
???? ?? , find 
????
????
 
Solution :  ?????? = ?????? . ???? ( ?????? ?? ) 
On differentiating we get, 
1
?? ????
????
=
1
?? ???? ( ?????? ?? )+ ?????? .
?????? ?? ?????? ?? ?
????
????
= ( ?????? ?? )
???? ?? [
?????? ( ?????? ?? )
?? + ?????? ?????? ?? ]  Ans. 
Problem 6 : If ?? = ?????? (?????? -1
 (
?? -?? 2
?? 2
) ), then 
????
????
 equals - 
(A) ?? [1 + ?????? ( ?????? ?? )+ ?????? 2
 ?? ] 
(B) 2?? [1 + ?????? ( ?????? ?? ) ] + ?????? 2
 ?? 
(C) 2?? [1 + ?????? ( ?????? ?? ) ] + ?????? ?? 
(D) 2?? + ?? [1 + ?????? ( ?????? ?? ) ]
2
 
Solution : Taking log on both sides, we get 
 ?????? ?? = ?????? -1
 (
?? - ?? 2
?? 2
) ? ?????? ( ?????? ?? )= ( ?? - ?? 2
) /?? 2
   ? ?? = ?? 2
+ ?? 2
?????? ( ?????? ?? )   ???? ?????????????????????????????? , ???? ??????   
? 
????
????
= 2?? + 2???????? ( ?????? ?? )+ ?? ?????? 2
 ( ?????? ?? )? 2?? [1 + ?????? ( ?????? ?? ) ] + ?? ?????? 2
 ( ?????? ?? ) 
= 2?? + ?? [1 + ?????? ( ?????? ?? ) ]
2
 
Ans. (D) 
Problem 7: If ?? =
?? 1/2
( 1-2?? )
2/3
( 2-3?? )
3/4
( 3-4?? )
4/5
 find 
????
????
 
Solution: 
?????? =
1
2
???? ?? +
2
3
???? ( 1 - 2?? )-
3
4
???? ( 2 - 3?? )-
4
5
???? ( 3 - 4?? ) 
On differentiating we get, 
? 
1
?? ????
????
=
1
2?? -
4
3( 1 - 2?? )
+
9
4( 2 - 3?? )
+
16
5( 3 - 4?? )
 
????
????
= ?? (
1
2?? -
4
3( 1 - 2?? )
+
9
4( 2 - 3?? )
+
16
5( 3 - 4?? )
) 
5. PARAMETRIC DIFFERENTIATION : 
If ?? = ?? ( ?? ) & ?? = ?? ( ?? ) where ?? is a parameter, then 
????
????
=
???? /????
???? /????
=
?? '
( ?? )
?? '
( ?? )
 
Problem 8: If ?? = ???????? ?? and ?? = ?? ( ?? - sint ) find the value of 
????
????
 at ?? =
?? 2
 
Solution :  
????
????
=
-???????? ?? ?? ( 1-?????? ?? )
?
????
????
|
?? =
?? 2
= -1 
Problem 9: Prove that the function represented parametrically by the equations. ?? =
1+?? ?? 3
; ?? =
3
2?? 2
+
2
?? satisfies the relationship : ?? ( ?? '
)
3
= 1 + ?? '
 ( where ?? '
=
????
????
 ) 
Solution : 
Here ?? =
1+?? ?? 3
=
1
?? 3
+
1
?? 2
 
Differentiating w.r. to ?? 
????
????
= -
3
?? 4
-
2
?? 3
 
?? =
3
2?? 2
+
2
?? 
Differentiating w.r. to ??