UNIT 1
DIFFERENTIAL CALCULUS
It is the process of differentiating the given function simultaneously many times and the result obtained are called successive derivative.
Let be a differentiable function.
First derivative is denoted by
Second derivative is
Third derivative is
Similarly the nth derivative is
Example:
Function  Derivaties 
…………..  
………..  
…………  
………….

Example1: Find the nth derivative of
Since
Differentiating both side with respect to x
[
Again differentiating with respect to x
Again differentiating with respect to x
Similarly the nth derivative is
Example2: Find the nth derivative of
Let
]
Differentiating with respect to x we get
Again differentiating with respect to x we get
Again differentiating with respect to x we get
Similarly Again differentiating with respect to x we get
Example3: Find the nth derivative
Let
Differentiating with respect to x.
Again differentiating with respect to x.
Again differentiating with respect to x.
Again differentiating with respect to x.
Again differentiating with respect to x.
Similarly the nth derivative with respect to x.
If u and v are the function of x such that their nth derivative exists, then the nth derivative of their product will be
Example1:Find the nth derivative of
Let
Also
By Leibnitz’s theorem
…(i)
Here
Differentiating with respect to x, we get
Again differentiating with respect to x, we get
Similarly the nth derivative will be
From (i) and (ii) we have,
Example2: If , then show that
Also, find
Here
Differentiating with respect to x, we get
…(ii)
Squaring both side we get
…(iii)
Again differentiating with respect to x ,we get
Using Leibnitz’s theorem we get
…(iv)
Putting x=0 in equation (i),(ii) and (iii) we get
Putting n=1,2,3,4….
………………
Hence
Example3: If then show that
Given
Differentiating both side with respect to x.
…..(ii)
Again differentiating with respect to x, we get
…(iii)
By Leibnitz’s theorem
…(iv)
Putting x=0 in equation (i),(ii),(iii) and (iv) we get
Putting n=1,2,3,4… so we get
Hence we have
Taylor’s theorem:
If (i) f(x) and its first (n1) derivative be continuous in [a, a+h],
(ii) exist for every value of x in (a, a+h), then there is at least one number such that
This is called Taylor’s theorem with Lagrange’s form of remainder
Taylor’s Series:
If can be expanded as an infinite series, then
If possesses derivative of all orders and the remainder .
Corollary: Taking and in equation (i) we get
Taking in above we get Maclaurin’s series.
Example1: Expand the polynomial in power of , by Taylor’s theorem.
Let .
Also
Then
Differentiating with respect to x.
Again differentiating with respect to x the above function.
Again differentiating with respect to x the above function.
Also the value of above functions at x=2 will be
By Taylor’s theorem
On substituting above values we get
Example2: Expand in power of
Let
Also
Differentiating f(x) with respect to x.
Again differentiating f(x) with respect to x.
Again differentiating f(x) with respect to x.
Also the value of above functions at x=1 will be
By Taylor’s theorem
On substituting above values we get
=
Example3:Expand in power of. Hence find the value of correct to four decimal places.
Let
and .
Differentiating with respect to x.
Again differentiating with respect to x.
Again differentiating with respect to x.
Again differentiating with respect to x.
Also the value of above functions at will be
By Taylor’s theorem
On substituting above values we get
At
.
Maclaurin’s theorem:
This is a particular case of Taylor’s theorem in which a=0 and h=x in Taylor’s theorem.
If f(x) can be expanded as an infinite series, then
Where the remainder is
Example1:Ifusing Taylor’s theorem, show that for .
Deduce that
Let then
Differentiating with respect to x.
.Then
Again differentiating with respect to x.
Then
Again differentiating with respect to x.
Then
By Maclaurin’s theorem
Substituting the above values we get
Since
Hence
Example2: Prove that
Let
Differentiating with respect to x.
Again differentiating with respect to x.
Again differentiating with respect to x.
Again differentiating with respect to x.
and so on.
Putting , in above derivatives we get
so on.
By Maclaurin’s theorem
+………
Substituting the above values we get
Example3:Prove that
Let
Differentiating above function with respect to x.
Again differentiating above function with respect to x.
Again differentiating above function with respect to x.
Again differentiating above function with respect to x.
Putting , in above derivatives we get
so on.
By Maclaurin’s theorem
+………
Substituting the above values we get
Example: Expansion of Some standard series
Let and
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
2.
Let and
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
3.
Let and
Differentiating above function with respect to x.
.
By Maclaurin’s theorem
+………
Substituting the above values we get
4.
Let and also
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
5.
Let and also
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
6.
Let and also
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
7.
Let
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
++………
+………
8.
Let
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
+………
+………
9.
Let
Differentiating above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
+………
+………)
10.
Let
Differentiating the above function with respect to x.
By Maclaurin’s theorem
+………
Substituting the above values we get
Let are functions of x only and have zero limit when .(where a is a certain point)
is called the indeterminate form. It means that does not exist.
The indeterminate form is distributed in the following form:
1. Form 0/0:
If then
This is called LHospital Rule.
In case
Then
Differentiation of numerator and denominator are done separately as many times as required.
Example1: Evaluate the limits of
As we can see that
Therefore
Using LHospital rule
Again using LHospital Rule
Again using LHospital Rule
Hence
Example2:Evaluate the limits of
Since
Therefore
Using LHospital rule
Again using LHospital Rule
Hence
Example3:Evaluate
Since form ….(i)
Consider , let
Taking log on both side
Differentiating both side with respect to x we get
Or
Or ….(ii)
Using LHospital Rule in (i) we have
[using (ii)]
=0/0 form
Again Using LHospital Rule
form
where
Again Using LHospital Rule
=
= form
Again using LHospital rule
=
Hence .
2. Form :
If then
This is called LHospital Rule.
In case
.
Then
Differentiation of numerator and denominator are done separately as many times as required. We use series and standard limits.
Example1: Find
Since form
Using LHospital Rule
=
= form
Again Using LHospital Rule
=
Hence
Example2: Find
Since form
Using LHospital Rule
= form
Again using LHospital Rule
= form
Again Using LHospital Rule
=
Hence .
Example3: Evaluate the limit
Since form
Therefore form
Using LHospital Rule
form
Again Using LHospital Rule
Hence
Then will be of the form 0/0 when .
And will be of the form when
b. Form : If then
Then
c. Forms : If form.
Let
Taking log on both side we get
This is solved by the above method then we have will give.
Example1: Evaluate the limit
Since form
Let
Taking log on both side we get
form
Using LHospital Rule
form
Again using LHospital Rule
=
=
Here
Hence .
Example2: Evaluate the limit
Since form
Let
Taking log on both side we get
form
Using LHospital Rule we get
Here
Hence.
Example3:Evaluate the limit
Since
Let y =
Taking log on both side we get
form
Using LHospital Rule we get
So,
Hence
Reference Books:
1. A text book of Applied Mathematics Volume I and II by J.N. Wartikar and P.N. Wartikar
2. Higher Engineering Mathematics by Dr. B. S. Grewal
3. Advanced Engineering Mathematics by H. K. Dass
4. Advanced Engineering Mathematics by Erwins Kreyszig