BMTC - 131
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
3
DIFFERENTIATION
Block Introduction 3
Notations and Symbols 4
UNIT 9
An Introduction to Differentiation 5
UNIT 10
Some More Derivatives 57
UNIT 11
Higher Order Derivatives 91
Miscellaneous Examples and Exercises 129
Appendix 1
Parametric Representation of Curves 147
Appendix 2
Partial Fractions 151
, Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
L.S.R. College, University of Delhi University of Delhi
Prof. Amber Habib
Shiv Nadar University Faculty Members
Gautam Buddha Nagar, U.P School of Sciences, IGNOU
Prof. M. S. Nathawat (Director)
Prof. S. A. Katre
University of Pune, Pune Dr. Deepika
Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Dr. Amit Kulshreshtha Prof. Sujatha Varma
IISER, Mohali Dr. S. Venkataraman
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert Committee and
the UGC-CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. Deepika
Shiv Nadar University School of Sciences
Gautam Buddha Nagar, U.P IGNOU, New Delhi
Dr. Malathy A. (Language Editor)
School of Humanities
IGNOU, New Delhi
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Also, to Sh.
Santosh Kumar Pal for the word processing and to Sh. S. S. Chauhan for preparing CRC of this
block. Parts of this block are based on the course material of the previous course Calculus
(MTE-01).
July, 2019
© Indira Gandhi National Open University, 2019
ISBN-978-93-89200-40-9
All right reserved. No part of this work may be reproduced in any form, by mimeograph or any other means,
without permission in writing from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses, may be obtained from the
University’s office at Maidan Garhi, New Delhi-110 068 and IGNOU website www.ignou.ac.in.
Printed and published on behalf of the Indira Gandhi National Open University, New Delhi by
Prof. M. S. Nathawat, Director, School of Sciences.
2
,BLOCK 3 DIFFERENTIATION
This is the third of the five blocks which you will be studying for the course “Calculus”. We
shall begin this block by defining derivatives of various functions which we discussed in
Block 2.
In Unit 9 we shall find derivatives of some standard functions using the definitions of
derivatives. We shall also discuss algebra of derivatives. In this unit we shall find that
continuity is necessary for a function to be differentiable.
In Unit 10 we shall continue our discussion of derivatives to find derivative of logarithmic,
exponential and hyperbolic functions. We shall also discuss other differentiation
techniques such as method of logarithmic differentiation and implicit differentiation.
In Unit 11 we shall find derivative of a derivative of a function and will extend our
discussion to higher order derivatives. We shall apply higher order derivatives to find
polynomial approximation.
In Unit 9 to 11, we have included a number of examples. Please go through them carefully
they will help you in a better understanding of the concepts discussed and will also serve
as a guideline in solving the exercises.
At the end of the block, you will find miscellaneous examples and exercises covering the
concepts you have studied across the units. Please solve the exercises on your own. At
the end of each unit, and after the miscellaneous exercises, we do provide some
solutions/answers to the exercises concerned. These are only as a support for you to be
able to check whether you have been able to solve the problem correctly or not. Please do
not look at these solutions till you have spent enough time on studying the unit and trying
all the exercises.
After the miscellaneous examples and exercises, you will find two appendices.
Appendix 1: Parametric Representation of curves
Appendix 2: Partial Fractions
A word about some signs used in the unit! Throughout each unit, you will find theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign .
To show the end of an example, we use ***. Further, equations that need to be referred to
are numbered sequentially within a unit, as are exercises and figures. E1, E2 etc. Denote
the exercises and Fig. 1, Fig. 2, etc. denote the figures.
3
, NOTATIONS AND SYMBOLS (used in Block 3)
w.r.t. with respect to
dy (1)
, y , y′, D( y) the first derivative of y w.r.t. x .
dx
d
(f ( x ), f ′( x ) the first derivative of f ( x ) w.r.t x .
dx
d 2 y ( 2)
, y , f ′′( x ) the second derivative of y or f ( x ) w.r.t. x .
dx 2
dn y (n ) (n )
, y , f (x) the nth derivative of y or f ( x ) w.r.t. x .
dx n
≈ is approximately equal to
Also, see the list of notations and symbols in Block 1 and Block 2.
4
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
3
DIFFERENTIATION
Block Introduction 3
Notations and Symbols 4
UNIT 9
An Introduction to Differentiation 5
UNIT 10
Some More Derivatives 57
UNIT 11
Higher Order Derivatives 91
Miscellaneous Examples and Exercises 129
Appendix 1
Parametric Representation of Curves 147
Appendix 2
Partial Fractions 151
, Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
L.S.R. College, University of Delhi University of Delhi
Prof. Amber Habib
Shiv Nadar University Faculty Members
Gautam Buddha Nagar, U.P School of Sciences, IGNOU
Prof. M. S. Nathawat (Director)
Prof. S. A. Katre
University of Pune, Pune Dr. Deepika
Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Dr. Amit Kulshreshtha Prof. Sujatha Varma
IISER, Mohali Dr. S. Venkataraman
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert Committee and
the UGC-CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. Deepika
Shiv Nadar University School of Sciences
Gautam Buddha Nagar, U.P IGNOU, New Delhi
Dr. Malathy A. (Language Editor)
School of Humanities
IGNOU, New Delhi
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Also, to Sh.
Santosh Kumar Pal for the word processing and to Sh. S. S. Chauhan for preparing CRC of this
block. Parts of this block are based on the course material of the previous course Calculus
(MTE-01).
July, 2019
© Indira Gandhi National Open University, 2019
ISBN-978-93-89200-40-9
All right reserved. No part of this work may be reproduced in any form, by mimeograph or any other means,
without permission in writing from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses, may be obtained from the
University’s office at Maidan Garhi, New Delhi-110 068 and IGNOU website www.ignou.ac.in.
Printed and published on behalf of the Indira Gandhi National Open University, New Delhi by
Prof. M. S. Nathawat, Director, School of Sciences.
2
,BLOCK 3 DIFFERENTIATION
This is the third of the five blocks which you will be studying for the course “Calculus”. We
shall begin this block by defining derivatives of various functions which we discussed in
Block 2.
In Unit 9 we shall find derivatives of some standard functions using the definitions of
derivatives. We shall also discuss algebra of derivatives. In this unit we shall find that
continuity is necessary for a function to be differentiable.
In Unit 10 we shall continue our discussion of derivatives to find derivative of logarithmic,
exponential and hyperbolic functions. We shall also discuss other differentiation
techniques such as method of logarithmic differentiation and implicit differentiation.
In Unit 11 we shall find derivative of a derivative of a function and will extend our
discussion to higher order derivatives. We shall apply higher order derivatives to find
polynomial approximation.
In Unit 9 to 11, we have included a number of examples. Please go through them carefully
they will help you in a better understanding of the concepts discussed and will also serve
as a guideline in solving the exercises.
At the end of the block, you will find miscellaneous examples and exercises covering the
concepts you have studied across the units. Please solve the exercises on your own. At
the end of each unit, and after the miscellaneous exercises, we do provide some
solutions/answers to the exercises concerned. These are only as a support for you to be
able to check whether you have been able to solve the problem correctly or not. Please do
not look at these solutions till you have spent enough time on studying the unit and trying
all the exercises.
After the miscellaneous examples and exercises, you will find two appendices.
Appendix 1: Parametric Representation of curves
Appendix 2: Partial Fractions
A word about some signs used in the unit! Throughout each unit, you will find theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign .
To show the end of an example, we use ***. Further, equations that need to be referred to
are numbered sequentially within a unit, as are exercises and figures. E1, E2 etc. Denote
the exercises and Fig. 1, Fig. 2, etc. denote the figures.
3
, NOTATIONS AND SYMBOLS (used in Block 3)
w.r.t. with respect to
dy (1)
, y , y′, D( y) the first derivative of y w.r.t. x .
dx
d
(f ( x ), f ′( x ) the first derivative of f ( x ) w.r.t x .
dx
d 2 y ( 2)
, y , f ′′( x ) the second derivative of y or f ( x ) w.r.t. x .
dx 2
dn y (n ) (n )
, y , f (x) the nth derivative of y or f ( x ) w.r.t. x .
dx n
≈ is approximately equal to
Also, see the list of notations and symbols in Block 1 and Block 2.
4
