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Study Materials for 1st Year B.Tech Students Paper Name: Mathematics Paper Code : M101 Teacher Name: Amalendu Singha Mahapatra Chapter – 2

Successive Differentiation Lecture 3: Objective: In this section you will learn the following Definition of successive derivatives. The notion of successive differentiation. The Leibniz’s formula. The related problems. Definition of successive derivatives We have seen that the derivative of a function of is in general also a function of . This new function may also be differentiable, in which case the derivative of the first derivative is called the second derivative of the original function. Similarly, the derivative of the second derivative is called the third derivative; and so on to the -th derivative. Thus, if

and so on. Notation The symbols for the successive derivatives are usually abbreviated as follows:

If y = f(x), the successive derivatives are also denoted by

or or,

The

-th derivative

For certain functions a general expression involving may be found for the -th derivative. The usual plan is to find a number of the first successive derivatives, as many as may be necessary to discover their law of formation, and then by induction write down the -th derivative. Example 1. Given y = e ax , find

,

dn y . dx n

, ...,

.

Example 2. Given y = log x , find

,

,

dny . dx n

,

Example 3. Given y = sinx, find

,

dny . dx n

, ...

.

Assignment:

{

}

(1) If u = sinax + cosax , then show that vn = a n 1 + ( −1) n sin 2ax

1 2

.

WBUT 03

n n (2) If u = x 2n show that un = 2 { 1.3.5......... ( 2n − 1) } x , n being a positive integer. (3) If y = 2 cosx(sinx - cosx) then show that n +1 2 ( − 1) 2n ifniseven ( yn ) 0 = n −1 (−1) 2 2n ifnisodd

Objective type questions: (1) The nth derivative of (ax + b)10 when n >0 is (a) a10 (b) 10! a10 (c) 0 (d) 10! (2) If y = e − x , then yn is (a) e − x (b) (−1) n (c) (−1) n e − x (d) none. (3) If y = xsinx , then yn = π (a) x sin(n + x ) 2 π π (b) sin( x + x) + n sin((n − 1) + x) 2 2 π π (c) x sin(n + x) + sin((n − 1) + x) 2 2 (d) none.

Lecture 4: Objective:

WBUT 07

Leibnitz's Formula for the

-th derivative of a product :

This formula expresses the -th derivative of the product of two variables in terms of the variables themselves and their successive derivatives. If u and v are functions of , each possessing derivatives upto nth order, then the nth derivative of their product is given by (uv) n = un v + n c1un −1v1 + n c2 un −2 v2 + ...... + n cr un −r vr + .... + uvn , Where the suffixes of u and v denote the order of differentiations of u and v with respect to x. Example 1. Given y = e x log x , find

Solution. Let ,

, and

v = log x

d3y by Leibnitz's Formula. dx 3

; then

,

,

,

,

.

Substituting in the above theorem we get

Example 2. Given y = x 2 e ax , find by

Solution. Let , theorem we get

Assignment:

, and ,

dny Leibnitz's Formula. dx n

; then , ...,

,

, ,

, . Substituting in the

1. If y = ( x 2 − 1) n , then show that ( x 2 − 1) yn + 2 + 2 xyn +1 − n(n + 1) yn = 0 Hence deduce that d 2 dpn (1 − x ) + n(n + 1) pn = 0 dx dx dn 2 wherepn = n ( x − 1) n dx 2.If y = tan −1 x then show that (1 + x 2 ) yn +1 + 2nxyn + n(n − 1) yn −1 = 0 .Hence find ( yn )o m sin −1 x

3. If y = e show that (1 − x 2 ) yn + 2 − (2n + 1) xyn+1 − (n2 + m2 ) yn = 0 . Also find ( yn )o .

Objective type questions: 1. If y = sin 2x + cos 2x , then ( yn )o is equal to π π (a) (−1) n 2n (b) 2n (c) sin n + cos n (d) none 2 2 100 2. If y = 5.x + 3, theny100 = (a) 100! (b) 5×100! (c) 0 (d) none.

WBUT2006

WBUT03,05

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Successive Differentiation Lecture 3: Objective: In this section you will learn the following Definition of successive derivatives. The notion of successive differentiation. The Leibniz’s formula. The related problems. Definition of successive derivatives We have seen that the derivative of a function of is in general also a function of . This new function may also be differentiable, in which case the derivative of the first derivative is called the second derivative of the original function. Similarly, the derivative of the second derivative is called the third derivative; and so on to the -th derivative. Thus, if

and so on. Notation The symbols for the successive derivatives are usually abbreviated as follows:

If y = f(x), the successive derivatives are also denoted by

or or,

The

-th derivative

For certain functions a general expression involving may be found for the -th derivative. The usual plan is to find a number of the first successive derivatives, as many as may be necessary to discover their law of formation, and then by induction write down the -th derivative. Example 1. Given y = e ax , find

,

dn y . dx n

, ...,

.

Example 2. Given y = log x , find

,

,

dny . dx n

,

Example 3. Given y = sinx, find

,

dny . dx n

, ...

.

Assignment:

{

}

(1) If u = sinax + cosax , then show that vn = a n 1 + ( −1) n sin 2ax

1 2

.

WBUT 03

n n (2) If u = x 2n show that un = 2 { 1.3.5......... ( 2n − 1) } x , n being a positive integer. (3) If y = 2 cosx(sinx - cosx) then show that n +1 2 ( − 1) 2n ifniseven ( yn ) 0 = n −1 (−1) 2 2n ifnisodd

Objective type questions: (1) The nth derivative of (ax + b)10 when n >0 is (a) a10 (b) 10! a10 (c) 0 (d) 10! (2) If y = e − x , then yn is (a) e − x (b) (−1) n (c) (−1) n e − x (d) none. (3) If y = xsinx , then yn = π (a) x sin(n + x ) 2 π π (b) sin( x + x) + n sin((n − 1) + x) 2 2 π π (c) x sin(n + x) + sin((n − 1) + x) 2 2 (d) none.

Lecture 4: Objective:

WBUT 07

Leibnitz's Formula for the

-th derivative of a product :

This formula expresses the -th derivative of the product of two variables in terms of the variables themselves and their successive derivatives. If u and v are functions of , each possessing derivatives upto nth order, then the nth derivative of their product is given by (uv) n = un v + n c1un −1v1 + n c2 un −2 v2 + ...... + n cr un −r vr + .... + uvn , Where the suffixes of u and v denote the order of differentiations of u and v with respect to x. Example 1. Given y = e x log x , find

Solution. Let ,

, and

v = log x

d3y by Leibnitz's Formula. dx 3

; then

,

,

,

,

.

Substituting in the above theorem we get

Example 2. Given y = x 2 e ax , find by

Solution. Let , theorem we get

Assignment:

, and ,

dny Leibnitz's Formula. dx n

; then , ...,

,

, ,

, . Substituting in the

1. If y = ( x 2 − 1) n , then show that ( x 2 − 1) yn + 2 + 2 xyn +1 − n(n + 1) yn = 0 Hence deduce that d 2 dpn (1 − x ) + n(n + 1) pn = 0 dx dx dn 2 wherepn = n ( x − 1) n dx 2.If y = tan −1 x then show that (1 + x 2 ) yn +1 + 2nxyn + n(n − 1) yn −1 = 0 .Hence find ( yn )o m sin −1 x

3. If y = e show that (1 − x 2 ) yn + 2 − (2n + 1) xyn+1 − (n2 + m2 ) yn = 0 . Also find ( yn )o .

Objective type questions: 1. If y = sin 2x + cos 2x , then ( yn )o is equal to π π (a) (−1) n 2n (b) 2n (c) sin n + cos n (d) none 2 2 100 2. If y = 5.x + 3, theny100 = (a) 100! (b) 5×100! (c) 0 (d) none.

WBUT2006

WBUT03,05

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