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CHAPTER 1

SUCCESSIVE DIFFERENTIATION AND LEIBNITZ’S THEOREM

1.1 Introduction

Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications. Let be a differentiable function and let its successive derivatives be denoted by .

Common notations of higher order Derivatives of

 1 st Derivative: or or or or 2 nd Derivative: or or or or ⋮ Derivative: or or or or 1.2 Calculation of n th Derivatives i. Derivative of Let y = ⋮ ii. Derivative of Let y = , is a ⋮ iii. Derivative of

Let iv. Derivative of

Let

Similarly if

v. Derivative of

Let

Similarly

where

Similarly if

and

Putting Summary of Results Function
Derivative
y
=
=
y
=
=
=
=
y
=
y
=
Example 1 Find the
derivative of
Solution: Let

Resolving into partial fractions =
=
=
=
!
 Example 2 Find the derivative of Solution: Let = (sin10 + cos2 ) ∴ = Example 3 Find Let y = derivative of Solution:

Example 4

Solution:

Example 5

Solution:

=

=

=

=

=

Find the

Let

Find the

Let

Now

derivative of

=

derivative of Example 6 If , prove that Solution:

= =

=

=

=

=

Example 7

Solution:

Find the

Let

derivative of

and

 ⇒ = = = = =

Differentiating above

times w.r.t. x, we get

Substituting such that

Using De Moivre’s theorem, we get where
Example 8 Find the
derivative of
Solution: Let
=
where
=
and
=

Resolving into partial fractions

Differentiating

times w.r.t. , we get Substituting such that

Using De Moivre’s theorem, we get where

Example 9 If Solution: Example 10
If
Solution:
=

,

show that

=

,

show that

( )

= 1

Differentiating both sides w.r.t.

, we get

( )

+ = 0

1. derivative of

Find the

2. derivative of Ans.

Find the Exercise 1 A
Ans.

If

3. ,

4.

5.

If

If

6.

If

,

show that

, find

, show that

i.e. the

derivative of Ans.
, find
Ans.

i.e. the

where

derivative of

7. Find

8.

9.

If y =

If

differential coefficient of =

Ans.

, show that

=

, show that

=

1.2 LEIBNITZ'S THEOREM If and are functions of such that their derivatives exist, then the derivative of their product is given by

where

and

represent

derivatives of

and

respectively.

Example11

Find the

derivative of

 Solution: Let and Then and

By Leibnitz’s theorem, we have Example 12 Find the derivative of Solution: Let and

Then By Leibnitz’s theorem, we have Example 13 If , show that = 0 Solution: Here ⇒

Differentiating both sides w.r.t.

=

, we get

 ⇒ Using L z’ s theorem, we get ⇒ ⇒

Example 14

If

Prove that

Solution:

)

Differentiating both sides w.r.t.

, we get

Using Leibnitz’s theorem

Example 15 If

Solution:

Here

, show that

. Also find

 ⇒ ……② ⇒ ⇒ ⇒ = ……③

Usi

Putting

(1- Differentiating w.r.t.

, we get

L

z’

h

r

, we get

in ,and and

……

 Putting in ④ Putting in the above equation, we get

=

= 0

=

 ⇒ Example 16 If show that

Solution:

Here

……

= Differentiating above equation w.r.t.

Differentiating above equation

times w.r.t.

, we get

……

u

L

. Also find

z’

h

r

w

.

 ⇒ ⇒ ……④ To find Putting in ①, ②and ③ Also putting and in ,we get Putting in the above equation, we get

=

Example 17 If

, show that

. Also find

 Solution: Here …① ⇒ ……②

Differentiating equation

……

times w.r.t.

u

L

z’

theorem

To find

Also putting

……in , and , we get

Putting and in ,we get

Putting

in the above equation, we get

=

=

=

0

Example18

Solution:

If

Here and

show that

……

Squaring both the sides, we get

Differentiating the above equation w.r.t.

Differentiating the above equation

……

times w.r.t.

Also find

, we get

u

L

z’

To find

Putting

and

……

in , and , we get

h

r

w

Also putting

Putting

=

in ,we get

in the above equation, we get

=

= 0

1 .Find

Ans.

2. Find

, if

, if

3. If

4. If

Exercise 1 B

, prove that

Ans.

), prove that

5. If

6 If

7. If

Ans. , prove that

show that

and

, show that

. Also find

.

8.

If

prove that
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