Study Materials for 1st Year B.Tech StudentsPaper Name: Mathematics

Paper Code : M101Teacher Name: Amalendu Singha Mahapatra

Chapter – 2Successive Differentiation

Lecture 3:

Objective:

In this section you will learn the following

(1) Definition of successive derivatives.(2) The notion of successive differentiation.

(3) The Leibniz’s formula.

(4) 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 , find .

, , ..., .

Example 2.   Given y = log x , find .

, , , , ... .

Example 3.   Given y = sinx, find .

,

Assignment:

(1) If u = sinax + cosax , then show that . WBUT 03

(2) If show that , n being a positive integer.

(3) If y = 2 cosx(sinx - cosx) then show that

Objective type questions:

(1) The nth derivative of when n >0 is

(a) (b) 10! (c) 0 (d) 10! WBUT 07

(2) If , then is

(a) (b) (c) (d) none.

(3) If y = xsinx , then

(a)

(b)

(c)

(d) none.

Lecture 4:Objective: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

Where the suffixes of u and v denote the order of differentiations of u and v with respect to x.

Example 1.   Given , find by Leibnitz's Formula.

Solution. Let , and ; then , , , ,

, .

Substituting in the above theorem we get

Example 2.   Given , find by Leibnitz's Formula.

Solution. Let , and ; then , , ,

, , , ..., , . Substituting in the theorem we get

Assignment:

1. If , then show that

Hence deduce that

WBUT2006

2.If then show that

.Hence find WBUT03,05

3. If show that

.

Also find .

Objective type questions:

1. If y = sin 2x + cos 2x , then is equal to

(a) (b) (c) (d) none

2. If (a) 100! (b) 5×100! (c) 0 (d) none.