A TEXTBOOK OF ENGINEERING MATHEMATICS

By U. HIENL

CONTENTS CHAPTER PRELIMINARIES 0 1 LIMIT 0 2 ONE SIDED LIMITS 0 2 1 CLOSED INTERVAL 0 2 2 OPEN INTERVAL 0 2 3 ABSOLUTE VALUE 0 2 4 NEIGHBOURHOOD OF A POINT 0 3 THE DEFINITION OF LIMIT (CAUCHY’S DEFINITION) 0 4 ONE-SIDED LIMITS 0 4 1 LEFT-HAND LIMIT 0 4 2 RIGHT HAND LIMIT 0 5 ALGEBRA OF LIMITS 0 6 LIMITS AT INFINITY 0 7 CONTINUITY 0 8 DERIVATIVE OF A FUNCTION 0 9 SOME STANDARD RESULTS 0 9 1 THE DERIVATIVE OF A CONSTANT FUNCTION IS ZERO 0 10 DERIVABILITY AND CONTINUITY 0 11 SOME STANDARD RULES OF DIFFERENTIATION THEOREM DIFFERENTIATION OF SUM AND DIFFERENCE PRODUCT RULE QUOTIENT RULE CHAIN RULE DERIVATIVES OF INVERSE TRIGONOMETRICAL FUNCTION DIFFERENTIATION OF FUNCTIONS IN PARAMETRIC FORM IMPLICIT DIFFERENTIATION LOGARITHMIC DIFFERENTIATION DIFFERENTIATION OF ONE FUNCTION WITH RESPECT TO ANOTHER FUNCTION 0 12 HYPERBOLIC FUNCTIONS AND THEIR DERIVATIVES 0 13 HIGHER ORDER DERIVATIVES AND SUCCESSIVE DIFFERENTIATION 0 14 PARTIAL DIFFERENTIATION PARTIAL DERIVATIVES HOMOGENOUS FUNCTIONS 0 15 MAXIMA AND MINIMA INCREASING FUNCTION DECREASING FUNCTION STATIONARY POINT STATIONARY VALUE MAXIMUM AND MINIMUM VALUES OF A FUNCTION POINTS OF INFLEXION FIRST DERIVATIVE TEST SECOND DERIVATIVE TEST 0 16 INTRODUCTION 0 17 INDEFINITE INTEGRAL 0 18 STANDARD INTEGRATION FORMULAE 0 19 BASIC RULES OF INTEGRATION 0 20 METHODS OF INTEGRATION 0 20 1 INTEGRATION BY METHOD OF SUBSTITUTION 0 20 2 STANDARD FORMS 0 21 INTEGRATION BY PARTS 0 22 INTEGRALS OF THE TYPE 0 23 INTEGRALS OF THE A - X DX 0 24 INTEGRALS OF THE FORM X – A DX, A + X DX 0 25 DEFINITE INTEGRALS 0 26 PROPERTIES OF DEFINITE INTEGRAL 0 27 INTEGRATION BY PARTS CHAPTER MATRICES 1 1 INTRODUCTION 1 2 ELEMENTARY ROW AND COLUMN OPERATIONS 1 2 1 EQUIVALENT MATRICES 1 2 2 ELEMENTARY MATRICES 1 3 RANK OF A MATRIX 1 3 1 RANK AND ELEMENTARY MATRICES DIFFERENTIAL CALCULUS - I 2 1 INTRODUCTION 2 2 HIGHER ORDER DERIVATIVES 2 3 DERIVATIVES OF HIGHER ORDER AND LEIBNITZ’S THEOREM 2 3 1 LEIBNITZ’S THEOREM 2 4 PARTIAL DIFFERENTIATION 2 5 HOMOGENEOUS FUNCTIONS 2 6 EULER’S THEOREM 2 7 CHANGE OF VARIABLES 2 8 EXPANSIONS OF FUNCTIONS 2 8 1 TAYLOR’S THEOREM WITH LAGRANGE’S FORM OF REMAINDER 2 8 2 TAYLOR’S THEOREM WITH CAUCHY’S FORM OF REMAINDER 2 8 3 TAYLOR’S THEOREM WITH GENERALIZED FORM OF REMAINDER 2 8 4 MACLAURIN’S THEOREM WITH GENERALIZED FORM OF REMAINDER 2 8 5 TAYLOR’S THEOREM FOR THE FUNCTION OF TWO VARIABLES DIFFERENTIAL CALCULUS – II 3 1 INTRODUCTION 3 2 PROPERTIES OF JACOBIAN 3 3 CHAIN RULE 3 4 FUNCTIONAL DEPENDENCE 3 5 ERRORS AND APPROXIMATIONS 3 5 1 APPROXIMATE CALCULATIONS MULTIPLE INTEGRALS 4 1 INTRODUCTION 4 2 DOUBLE INTEGRALS 4 3 EVALUATION OF DOUBLE INTEGRALS IN POLAR COORDINATES 4 4 TRIPLE INTEGRALS 4 4 1 EVALUATION OF TRIPLE INTEGRALS 4 5 CHANGE OF VARIABLES 4 5 1 DOUBLE INTEGRALS 4 5 2 TRIPLE INTEGRALS 4 6 CHANGING THE ORDER OF INTEGRATION 4 7 APPLICATION OF MULTIPLE INTEGRALS MULTIPLE INTEGRALS 5 1 INTRODUCTION 5 2 POINT FUNCTIONS 5 2 1 SCALAR POINT FUNCTION 5 2 2 SCALAR FIELD 5 2 3 VECTOR POINT FUNCTION 5 2 4 VECTOR FIELD 5 3 VECTOR DIFFERENTIAL OPERATOR 5 3 1 LEVEL SURFACE 5 4 GRADIENT OF A SCALAR FUNCTION 5 4 1 MAGNITUDE OF 5 5 PROPERTIES OF GRADIENT 5 6 DIRECTIONAL DERIVATIVE 5 7 EQUATION OF TANGENT PLANE AND NORMAL TO A LEVEL SURFACE 5 7 1 EQUATION OF THE TANGENT PLANE 5 7 2 EQUATION OF THE NORMAL DIVERGENCE 5 9 CURL 5 9 1 IRROTATIONAL DIFFERENTIAL EQUATIONS 6 1 INTRODUCTION 6 2 SOLUTION OF A DIFFERENTIAL EQUATION 6 2 1 GENERAL SOLUTION 6 2 2 PARTICULAR SOLUTION 6 2 3 SINGULAR SOLUTION 6 3 FORMATION OF DIFFERENTIAL EQUATION 6 4 FIRST ORDER AND FIRST DEGREE DIFFERENTIAL EQUATIONS 6 5 VARIABLES CONTENTS 6 6 EQUATIONS REDUCIBLE TO THE FORM IN WHICH VARIABLES 6 7 HOMOGENOUS EQUATIONS 6 7 1 METHOD OF SOLVING HOMOGENOUS DIFFERENTIAL EQUATIONS 6 7 2 EQUATIONS REDUCIBLE TO HOMOGENOUS FORM 6 8 EXACT DIFFERENTIAL EQUATIONS 6 9 INTEGRATING FACTORS 6 9 1 NUMBER OF INTEGRATING FACTORS 6 10 RULES FOR FINDING THE INTEGRATING FACTORS PRELIMINARIES MANY PHENOMENA INVOLVE CHANGING QUANTITIES AND THEIR RATES OF CHANGE CALCULUS IS CONCERNED WITH THE MATHEMATICAL STUDY OF CHANGE THE FOUNDATIONS OF CALCULUS WERE LAID BY BRITISH MATHEMATICIAN AND PHYSICIST SIR ISAAC NEWTON (1642-1727) AND GERMAN MATHEMATICIAN GOTTFRIED WILHELM LEIBNITZ (1646-1716) THE SUBJECT OF DIFFERENTIAL CALCULUS IS ESSENTIALLY CONCERNED WITH THE RATE OF CHANGE OF A DEPENDENT VARIABLE WITH RESPECT TO AN INDEPENDENT VARIABLE THE CONCEPT OF LIMIT, CONTINUITY AND DIFFERENTIABILITY ARE THE BASIC TOOLS OF DIFFERENTIAL CALCULUS 0 1 LIMIT CALCULUS IS BASED ON THE CONCEPT OF LIMIT AND FUNCTION THE CONCEPT OF LIMIT HELPS TO DESCRIBE THE BEHAVIOUR OF A FUNCTION F(X) WHEN THE INDEPENDENT VARIABLE X TAKES VALUES VERY CLOSE TO A PARTICULAR WE NOW DISCUSS THE IDEA OF LIMIT OF A FUNCTION TO INTRODUCE THE IDEAL OF LIMIT OF A FUNCTION, WE CONSIDER THE FOLLOWING EXAMPLE EXAMPLE 1: CONSIDER F(X) = X X , X HERE THE FUNCTION F(X) DEFINED FOR ALL REAL VALUES OF X EXCEPT AT THE POINT X = 1 DIVIDING THE NUMERATOR AND DENOMINATOR BY (X – 1) WE OBTAIN F(X) = 3 (X + 1), X NOW WE COMPUTE THE VALUES OF F(X) FOR SOME VALUE OF X NEAR 1, BUT NOT EQUAL TO 1, AND OBTAIN THE FOLLOWING TABLES …… F(X) 3(2 1) 3(2 01) 3(2 001) 3(2 0001) …… WE OBSERVE FROM TABLE 0 1 THAT WHEN X GETS CLOSER AND CLOSER TO 1, THROUGH THE VALUES LESS THAN 1, THE FUNCTION F(X) BECOMES CLOSER TO AND CLOSER TO 3 2 = 6 SIMILARLY IN TABLE 0 2 WHEN X ASSUMES VALUES NEARER AND NEARER TO 1, F(X) BECOMES CLOSER TO 6 IT IS ALSO EVIDENT THAT WE CAN MAKE, F(X) AS NEAR TO 6 AS WE DESIRE BY TAKING X SUFFICIENTLY NEAR TO 1 IN SUCH A SITUATION WE SAY THAT THE LIMIT OF F(X), AS X APPROACHES 1 IS 6 AND WRITE: 1 ( ) XLT F X IN GENERAL, SUPPOSE A FUNCTION F IS DEFINED FOR X NEAR A BUT NOT NECESSARILY AT X = A IF L BE A FIXED NUMBER SUCH THAT F(X) CAN BE MADE AS CLOSE TO L AS WE DESIRE BY CHOOSING X SUFFICIENTLY CLOSE TO A, THEN WE SAY THAT L IS THE LIMIT OF F(X) AS X APPROACHES A AND DENOTE IT AS: ( ) X A LT F X L EXAMPLE 2: IF X F X LT F X X DOES NOT EXIST SOLUTION: AS X GETS NEARER AND NEARER TO 3 FROM EITHER SIDE, F(X) BECOMES ARBITRARILY LARGE THUS F(X) DOES NOT APPROACH ANY FIXED NUMBER AS X APPROACHES 3 THEREFORE ( ) XLT F X DOES NOT EXIST 0 2 ONE SIDED LIMITS BEFORE EXPLAINING THESE CONCEPTS WE DEFINE THE FOLLOWING 0 2 1 CLOSED INTERVAL IF A AND B ARE TWO REAL NUMBERS (A < B), THEN THE SET {X R | A < X < B} IS CALLED A CLOSED INTERVAL FROM A TO B, IT IS DENOTED BY [A, B] [A, B] = {X | A < X < B} A TEXTBOOK OF ENGINEERING MATHEMATICS 0 2 2 OPEN INTERVAL IF A AND B ARE TWO REAL NUMBERS (WHERE A < B), THEN THE SET {X R | A < X < B} IS CALLED AN OPEN INTERVAL FROM A TO B THE OPEN INTERVAL OF A AND B IS DENOTED BY (A, B) (A, B) = {X R | A < X < B} 0 2 3 ABSOLUTE VALUE THE ABSOLUTE VALUE OF A REAL NUMBER X, DENOTED BY |X| (READ AS MODULUS OF X) IS DEFINED AS | X | = X, IF X > 0 AND | X | = –X, IF X < EXAMPLE 3: | 5 | = 5, | –4 | = – (–4) = ( ) X ALT F X L EXAMPLE: CONSIDER THE FUNCTION F(X) DEFINED AS FOLLOWS F(X) = 4 ; IF ; IF X X < X A TEXTBOOK OF ENGINEERING MATHEMATICS F IS DEFINED AS F(X) = 4X WHEN X < 0 BY THE DEFINITION; IF X IS IN THE LEFT NEIGHBOURHOOD OF ZERO I E , X < 0, WE HAVE – – X X LT F X LT X DEFINITION (RIGHT HAND LIMIT): LET A FUNCTION F BE DEFINED IN SOME RIGHT NEIGHBOURHOOD (A, A + ) OF A SUCH THAT F(X) IS ARBITRARILY CLOSE TO 'L WHENEVER X IS IN A SUFFICIENTLY SMALL RIGHT NEIGHBOURHOOD OF A THEN THE RIGHT LIMIT OF THE RIGHT HAND LIMIT OF F(X) AS X A IS 'L AND DENOTED AS ( ) ' X A LT F X L OR ( ) ' X ALT F X L EXAMPLE: CONSIDER THE FUNCTION F(X) DEFINED AS FOLLOWS 2; IF ( ) 2; IF X X < F X X X CLEARLY F(X) IS DEFINED IN THE RIGHT NEIGHBOURHOOD OF ZERO I E , FOR X > 0 IF X IS THE RIGHT NEIGHBOURHOOD OF X ZERO THEN F(X) = X + 2, HENCE XLT X SOLUTION: LET > 0 BE GIVEN WE SEEK A REAL NUMBER > 0 SUCH THAT A TEXTBOOK OF ENGINEERING MATHEMATICS XLT X 0 4 ONE-SIDED LIMITS 0 4 1 LEFT-HAND LIMIT LET A FUNCTION F BE DEFINED FOR VALUES X < A AND L BE A REAL NUMBER THEN THE LIMIT F(X) AS A APPROACHES A THROUGH THE VALUES LESS THAN A (I E , THE LEFT-HAND LIMIT L OF F(X)) IS DEFINED AS FOLLOWS: GIVEN > 0, HOWEVER SMALL, IF THERE EXISTS A > 0 SUCH THAT | F(X) – L | < WHENEVER 0 < A – X < , THEN L IS CALLED THE LEFT HAND LIMIT OF F(X) AS X APPROACHES A IT IS WRITTEN AS – ( ) X A LT F X L NOTE: TO FIND THE LEFT-HAND, REPLACE X BY A – H, (H > 0) AND MAKE H TEND TO ZERO 0 4 2 RIGHT HAND LIMIT LET F BE A FUNCTION DEFINED FOR VALUES OF X > A AND L BE A REAL NUMBER THE RIGHT HAND LIMIT L OF F(X) IS DEFINED AS FOLLOWS: GIVEN > 0, HOWEVER SMALL, IF THERE EXIST A > 0, SUCH THAT | F(X) – L | < WHENEVER 0 < X – A < , THEN WE SAY L IS THE RIGHT-HAND LIMIT OF F(X) AS X APPROACHES A IT IS WRITTEN AS ( ) X A LT F X L NOTE: TO FIND THE RIGHT-HAND LIMIT, REPLACE X BY A + H, AND MAKE H 0 A TEXTBOOK OF ENGINEERING MATHEMATICS 0 5 ALGEBRA OF LIMITS THE COMPUTATION OF LIMITS IS SIMPLIFIED BY USING CERTAIN PROPERTIES WHICH ARE STATED AS FOLLOWS IF F AND G ARE TWO FUNCTIONS SUCH THAT ( ) X A LT F X L OR ( ) X A LT G X M , THEN (I) [ ( ) ( )] ( ) ( ) X A X A X A LT F X G X LT F X LT G X L M (II) [ ( )] ( ) (K R) X A X A LT K F X K LT F X KL (III) [ ( ) ( )] ( ) ( ) X A X A X A LT F X G X LT F X LT G X L M (IV) ( ) ( ) ; ( ) ( ) X A X A X A LT F X F X L LT G X LT G X M IF ( ) – – N N N X A X A LT N A X A (N = A NON-ZERO RATIONAL NUMBER) (II) N XLT E X (III) 1/ 0 X XLT X E (IV) LOG X E X A LT A X (V) SIN X X LT X A TEXTBOOK OF ENGINEERING MATHEMATICS SOLVED EXAMPLES EXAMPLE 1: EVALUATE – X X LT X AND X X LT X AND PROVE THAT X X LT X DOES NOT EXITS SOLUTION: FROM THE DEFINITION; – ; IF X X > X X X – – XLT – X X X X LT LT X X 0 (1) XLT CLEARLY – X X X X LT LT X X – X X X X LT LT X X DOES NOT EXIST EXAMPLE 2: EVALUATE – XLT F X AND XLT F X AND SHOW THAT XLT F X EXISTS 2 –1; IF – 2 ; IF X X F X X X X> SOLUTION: – – X X LT F X LT X PUT X = 1 – H, X 1 H (H > 0) – [2(1– ) –1] H XLT X LT H 0(1– 2 ) HLT H = 1 – 0 = AND X X LT F X LT X X PUT X = 1 + H X 1 H A TEXTBOOK OF ENGINEERING MATHEMATICS )] H XLT X X LT H H = )] HLT H H H = 6 – 2 – 2 )] HLT H H H = XLT F X BOTH EXIST AND ARE EQUAL XLT F X EXAMPLE 3: EVALUATE X X LT X SOLUTION – – N N N X A X A LT N A X A ! " = 4 23 = 4 8 = EXAMPLE 4: EVALUATE XLT X X SOLUTION XLT X X = X X X LT X LT X LT = 3 12 + 5 1 + = 3 + 5 + 1 = EXAMPLE 5: THE CONSUMPTION EXPENDITURE E AS A FUNCTION OF PERSONAL INCOME Y IS GIVEN BY E = 20 + 0 8 Y FIND THE LIMIT OF SAVINGS S WHERE S = Y – E WHEN Y TENDS TO 300 SOLUTION: SAVINGS S = Y – E = Y – [20 + 0 8Y] = Y – 20 – 0 8 Y = 0 2 Y – Y Y LT S LT Y = 0 2 × 300 – 20 = 60 – 20 = EXAMPLE 6: FIND SOLUTION X X X + X X X LT LT X X + X X (DIVIDING THE NUMERATOR AND DENOMINATOR BY X) = X + X LT + X = X X EXAMPLE 7: IT IS KNOWN THAT INCOME, YT IS GROWING WITH TIME T AND IS GIVEN BY YT = 2 + 3T + 24T2 THE POPULATION NT IS KNOWN TO BE INCREASING WITH T AND IS GIVEN BY NT = 3+ 5T + 12T2 FIND THE LIMITING VALUE OF PER CAPITAL INCOME AS T TENDS TO SOLUTION: LET FT DENOTE THE PER CAPITAL INCOME AT TIME T T T T Y F N T T X T T Y LT F LT N T T T LT T T ! " T T T T T T LT T T T T T ! " T T T LT T T 0 AND 0) T T T THE LIMITING VALUE OF PER CAPITAL INCOME AS T , APPROACHES 2 A TEXTBOOK OF ENGINEERING MATHEMATICS EXAMPLE 8: FIND # X X X LT X SOLUTION: THE LIMIT OF THE DENOMINATOR IS ZERO WHEN # X WE FIND THE LIMIT OF THE FUNCTION AS FOLLOWS ( 1) ( 1) # # X X X X X X LT LT X X # # XLT X I EVALUATE THE FOLLOWING LIMITS 1 # X X X LT X X 2 # X X X LT X X 3 # # X X LT X 4 # # X X LT X 5 # # X X LT X 6 # # X X LT X 7 ( 4) # # X X X LT X 8 # # M M N N X A X A LT X A 9 # X X X LT X 10 # # X X X LT X II PROVE THE FOLLOWING (I) # X X X LT X X A TEXTBOOK OF ENGINEERING MATHEMATICS (II) # X X X LT X X (III) X X X LT X (IV) X N LT N (HINT: 1 + 2 + + ( 1) X X X LT X X III (I) F(X) IS DEFINED AS FOLLOWS 2; IF 3 ; IF # $ # X X F X X X X SHOW THAT XLT F X (II) LET # % X X F X X X SHOW THAT X LT F X ANSWERS I 1 8 – M N M A N 9 1/ 2 0 7 CONTINUITY DEFINITION: A FUNCTION F(X) IS SAID TO CONTINUOUS AT A POINT X = A; IF SHOW THAT X A LT F X F A NOTE: F(A) IS CALLED THE VALUE OF THE FUNCTION X = A FROM THE ABOVE DEFINITION IT IS CLEAR THAT, A FUNCTION F(X) IS CONTINUOUS AT X = A IF (I) F IS DEFINED AT THE POINT X = A (II) X A LT F X EXISTS, THAT IS A TEXTBOOK OF ENGINEERING MATHEMATICS – X A X A LT F X LT F X AND (III) X A LT F X F A A FUNCTION WHICH IS CONTINUOUS AT EVERY POINT OF AN INTERVAL IS SAID TO BE CONTINUOUS THROUGHOUT THAT INTERVAL A FUNCTION IS SAID TO BE DISCONTINUOUS AT A POINT IF IT IS NOT CONTINUOUS AT THAT POINT CONTINUOUS FUNCTIONS POSSESS THE FOLLOWING PROPERTIES; 1 THE SUM, THE DIFFERENCE AND THE PRODUCT OF FUNCTIONS CONTINUOUS AT A POINT ARE ALSO CONTINUOUS AT THAT POINT 2 THE QUOTIENT Y = F(X)/G(X) OF TWO FUNCTIONS CONTINUOUS AT THE POINT X = A, IS A FUNCTION WHICH IS CONTINUOUS AT THIS POINT G(A) 0 3 IF A FUNCTION WHICH IS CONTINUOUS ON A CLOSED INTERVAL ASSUMES VALUES OF DIFFERENT SIGNS AT ITS END POINTS THEN THERE IS ATLEAST ONE POINT ON THAT INTERVAL AT WHICH THE FUNCTION VANISHES SOLVED EXAMPLES EXAMPLE 1: A FUNCTION F IS DEFINED AS FOLLOWS 1; IF 1; IF $ # X X F X X X SHOW THAT F(X) IS CONTINUOUS AT X = 1( 1) X X LT F X LT X = 0[(1– ) 1] HLT H (PUTTING X = 1 – H, WE GET X 1 H 0 ) = 0(2 – ) 2 – HLT H – X LT F X 3 ) 2 – 2] HLT H A TEXTBOOK OF ENGINEERING MATHEMATICS X LT F X 3 –1; IF F X X X F X X LT F X LT F X F I E , XLT F X F F(X) IS CONTINUOUS AT X = EXAMPLE 2: A FUNCTION F IS DEFINED AS FOLLOWS ( – 4) / – 2 ; IF ( ) 4; IF & & X X X F X X SHOW THAT F(X) IS CONTINUOUS AT X = X X X LT F X LT X = X X X LT X = XLT X 1 SHOW THAT F(X) IS CONTINUOUS AT X = 2; WHERE ( ) 5 – ; IF $ X X F X X X 2 A FUNCTION F IS DEFINED AS FOLLOWS $ & & X X F X X X SHOW THAT F IS CONTINUOUS AT X = 3 TEST FOR THE CONTINUITY OF THE FUNCTION F(X) AT X = 1; WHERE 2 – – 6 ; 5 ; = & & X X X F X X X (ANS CONTINUOUS) 4 SHOW THAT F(X) = – (X – 1) IS DISCONTINUOUS AT X = 5 A FUNCTION F(X) IS DEFINED AS FOLLOWS 3 – 8; K ; = & & X X F X X X IF F(X) IS CONTINUOUS AT X = 2 FIND K (ANS K = 12) 0 8 DERIVATIVE OF A FUNCTION A FUNCTION F IS SAID TO BE DERIVABLE AT A POINT X IF THE LIMIT – X F X H F X LT H …(1) EXISTS THE LIMITING VALUE IS CALLED THE DERIVATIVE OR THE DIFFERENTIAL COEFFICIENT OF THE FUNCTION F AT X AND IS DENOTED BY D F(X) OR BY 'F X – ' X F X H F X D F X F X LT H …(2) (THE OPERATOR D = D DX ) THE PROCESS OF FINDING THE DERIVATIVE OF THE FUNCTION IS KNOWN AS DIFFERENTIATION THE DERIVATIVE OF F(X) IS ALSO CALLED THE DIFFERENTIAL CO-EFFICIENT OF F(X) OTHER NOTATIONS AND TERMINOLOGIES IN THE FUNCTIONAL WE USE DY/DX FOR THE DERIVATIVE OF THE FUNCTION Y = F(X) INSTEAD OF H THE NOTATION 'X IS ALSO USED, 'X IS CALLED THE INCREMENT OVEN X WHERE 'X < 0 OR > 0 THE CORRESPONDING CHANGE IN THE FUNCTION Y = F(X) IS WRITTEN AS 'Y USING THE NOTATION WE CAN WRITE – ' X X F X X F X DY Y F X LT LT DX X X ' ' ' ' ' ' DIFFERENTIATING A FUNCTION F IS REFERRED TO AS “DIFFERENTIATION WITH RESPECT TO X FROM THE FIRST PRINCIPLES” OR “DIFFERENTIATION WITH RESPECT TO X ABINITIO” A TEXTBOOK OF ENGINEERING MATHEMATICS THE NOTATIONS ', , Y Y F , DF/FX ARE ALSO USED TO DENOTE THE DERIVATIVE OF A FUNCTION F NOTE: A FUNCTION WHICH HAS A DERIVATIVE AT EVERY POINT OF AN INTERVAL IS SAID TO BE DIFFERENTIABLE ON THAT INTERVAL EXAMPLE 1: FIND THE DERIVATIVE OF 3X +1, FROM FIRST PRINCIPLES SOLUTION: LET F X X 3( ) F X X X X ' ' X X ' – F X X F X X X X X X ' ' ' ' X X X X ' ' X X ' ' X X F X X F X LT LT X ' ' ' ' F X OR LET Y X 3( ) 1) Y Y Y X X X ' ' X X X X ' ' 3 Y X ' ' Y X ' ' APPLYING LIMITS ON BOTH SIDES WE GET X X Y LT LT X ' ' ' ' A TEXTBOOK OF ENGINEERING MATHEMATICS DY DX I E , D X DX EXAMPLE 2: USING DEFINITION FIND THE DERIVATIVE OF X2 WITH RESPECT TO X SOLUTION: LET F X X WE GET 2 F X X X X X X X X ' ' ' ' – 2 – F X X F X X X X X X X X ' ' ' ' ' 2 X X X X ' ' ' ( 2 ) X X X X X X ' ' ' ' X X F X X F X LT LT X X X ' ' ' ' ' ' F X X OR – – Y X Y Y Y X X X ' ' ( ) 2 – Y X X X X X ' ' ' Y X X X ' ' ' X X Y LT LT X X X X ' ' ' ' ' DY X DX EXAMPLE 3: USE THE DEFINITION OF A DERIVATIVE OF A FUNCTION AND FIND THE DERIVATIVE OF X X WITH RESPECT X A TEXTBOOK OF ENGINEERING MATHEMATICS SOLUTION: LET X F X X WE GET X X X X F X X X X X X ' ' ' ' ' X X X F X X F X X X X ' ' ' 2 – X X X X X X X X X ' ' ' – 6 – 4 – X X X X X X X X X X X X X X X ' ' ' ' ' 2 – 2 X X F X X F X X LT LT X X X X X ' ' ' ' ' ' ' DIFFERENTIATE THE FOLLOWING FUNCTIONS FROM FIRST PRINCIPLES 1 X 2 X X 3 X X 4 X 5 X X 7 AX B CX D 8 2 2X + X 6 X 7 – AD BC CX D 8 X A TEXTBOOK OF ENGINEERING MATHEMATICS 0 9 SOME STANDARD RESULTS 0 9 1 THE DERIVATIVE OF A CONSTANT FUNCTION IS ZERO PROOF: LET F X C WHERE C IS A CONSTANT F(X + 'X) = C FORM THE DEFINITION OF A DERIVATIVE – ' X F X X F X F X LT X ' ' ' = X X C C LT LT X X ' ' ' ' EXAMPLES: (I) IF F(X) = 5 THEN ' F X (II) IF Y = – 4 THE DY/DX = PROOF: LET F(X) = XN N F X X X X ' ' – ' X F X X F X F X LT X ' ' ' DY Y X X X DX (III) F X X A TEXTBOOK OF ENGINEERING MATHEMATICS THE DERIVATIVES OF MANY FUNCTIONS CAN BE THUS OBTAINED “FROM THE FIRST PRINCIPLES” WE SHALL GIVE THE LIST OF STANDARD DERIVATIVES (SEE TABLE 6 3) FOR IMMEDIATE REFERENCE TABLE 6 3 DERIVATIVES OF SOME STANDARD FUNCTIONS Y = F X F' X (I E , DY/DX) C (CONSTANT) 2 X N X 1– X COS–1X 1– X TAN–1X 1 X COT–1X X SEC–1X X X COSEC–1X X X 0 10 DERIVABILITY AND CONTINUITY WE STATE THE FOLLOWING THEOREM WITHOUT PROOF A TEXTBOOK OF ENGINEERING MATHEMATICS THEOREM: IF A FUNCTION Y = F(X) IS DIFFERENTIABLE AT ANY POINT OF IT DOMAIN THEN IT IS CONTINUOUS AT THE POINT THE CONVERSE OF THE THEOREM IS NOT TRUE EVERY CONTINUOUS FUNCTION NEED NOT BE DIFFERENTIABLE THAT IS, IF F(X) IS CONTINUOUS AT A POINT, THEN IT MAY OR MAY NOT BE DIFFERENTIABLE AT THAT POINT FOR EXAMPLE, THE FUNCTION | |; 0, F F X X X IS CONTINUOUS AT X = 0, BUT NO DIFFERENTIABLE AT X = THEOREM THE DERIVATIVE OF A SCALARY MULTIPLE OF A FUNCTION IS THE SCALAR MULTIPLE OF THE DERIVATIVE OF THE FUNCTION PROOF: LET F X K F X WHERE K IS A CONSTANT WE GET F X X ' = ' K F X X – – ' ' F X X F X K F X X K F X – – X X F X X F X F X X F X LT LT K X X ' ' ' ' ' ! " DIFFERENTIATION OF SUM AND DIFFERENCE THEOREM: THE DERIVATIVE OF THE SUM (OR DIFFERENCE) OF THE DIFFERENTIABLE FUNCTIONS IS EQUAL TO THE CORRESPONDING SUM OF THEIR DERIVATIVES PROOF: LET F X F X G X , WHERE F AND G ARE DIFFERENTIABLE WE GET ' ' ' F X X F X X G X X – – – F X X F X F X X G X X F X G X ' ' ' A TEXTBOOK OF ENGINEERING MATHEMATICS – – – F X X F X F X X F X G X X G X X X X ' ' ' ' ' ' LT – – – X X F X X F X F X X F X LT X X G X X G X LT X ' ' ' ' ' ' ' ' ' ' F X F X G X SIMILARLY; IF – F X F X G X , WE CAN PROVE THAT ' ' – ' F X F X G X COROLLARY: THE DERIVATIVE OF THE LINEAR SUMS (OR DIFFERENCE) OF A FINITE NUMBER OF DIFFERENTIABLE FUNCTIONS IS EQUAL TO THE CORRESPONDING SUM OF THE DERIVATIVES OF THESE FUNCTIONS I E , IF, 1 2 N N F X A F X A F A F X ( ( THEN, 1 LOG X X X (III) LOG XE X X X (IV) 2 – LOG XE X X X SOLUTION: (I) LET F X X DIFFERENTIATING WITH RESPECT TO X, WE GET ' 7 7 X X X A TEXTBOOK OF ENGINEERING MATHEMATICS F X X X X (III) LET LOG F X E X X X DIFFERENTIATING WITH RESPECT TO X, WE GET ' LOG X D D D D F X E X X X DX DX DX DX X X E X X E X X (IV) LET 2 – LOG X F X E X X X DIFFERENTIATING WITH RESPECT TO X, WE GET ' – LOG X D D D D F X E X X X DX DX DX DX 2 –1/ XE X X EXAMPLE 2: IF 4 X X Y X X E A FIND DY/DX SOLUTION 4 X X Y X X E A DIFFERENTIATING WITH RESPECT TO X, WE GET X X DY D D D D X X E A DX DX DX DX DX LOG X X X E A A X EXAMPLE 3: DIFFERENTIATE THE FOLLOWING FUNCTIONS (I) X (II) X X X X (III) X X SOLUTION: (I) LET Y X THEN Y X A TEXTBOOK OF ENGINEERING MATHEMATICS –3 DY X X DX X (II) LET X X X Y X THEN – X X X Y X X X X = 4 / X X X DIFFERENTIATING WITH RESPECT TO X, WE GET DY X DX X 1– X X (III) LET Y X X X X DIFFERENTIATING WITH RESPECT TO X, WE GET 3 X X 2 3 X 3 X X [HINT X X X X ] 4 X X X [HINT X X X X X X ] 5 1– X X [HINT 1– 1 – 1– X X X X ] 6 X X X X [HINT – 1– X X X X X X X ] 7 X X X X X X X X ] A TEXTBOOK OF ENGINEERING MATHEMATICS 1– 1– X X X ] 9 LOG X [HINT LOG 4LOG X X ] II DIFFERENTIATE WITH RESPECT TO T; 1 Y AT 2 LOG Y T 3 Y T T III FIND DY/DX 1 1/ Y X 3 5 – X X X 5 X [HINT X X X X ] 6 X X X ] 7 – 3 X X X X X X X ] 8 X X X X X ] 9 X X X X ] 10 X X 11 – X X 12 X 3 2 / X X 7 X X L 8 –2X 9 4/X II 1 2 AT 2 1/T 3 3T2 + X III 1 –7 2 X 3 X X 5 X X 6 2 – X X 8 1 1/ X 9 2 – 2 / X X 10 1–1/ X 11 2 / X X 12 PRODUCT RULE THEOREM : IF F X AND G X ARE TWO DIFFERENTIABLE FUNCTIONS IN X, THEN ' ' ' F G X F X G X G X F X PROOF: LET F X F G X F X G X THEN F X X F X X G X X ' ' ' – – F X X F X F X X G X X F X G X ' ' ' – – F X X G X X F X X G X F X X G X F X G X ' ' ' ' ( ) [ ( ) – ( )] + ( ) [ ( + ) – ( )] ' ' ' F X X G X X G X G X F X X F X – – X X X F X X F X G X X G X LT LT F X X LT X X ' ' ' ' ' ' ' ' – X F X X F X G X LT X ' ' ' ' ' ' F X F X G X G X F X ' ' ' F X F X G X G X F X ALTERNATIVE NOTATION: THE ABOVE THEOREM CAN BE EXPRESSED AS FOLLOWS: IF U AND V ARE TWO DERIVABLE FUNCTIONS THEN D D D UV U V V U DX DX DX I E , ' ' ' UV UV UV PROOF: LET Y UV THEN Y Y U U V V ' ' ' + UV U V V U U V ' ' ' ' – – Y Y Y UV U V V U U Y UV ' ' ' ' ' A TEXTBOOK OF ENGINEERING MATHEMATICS Y U V V U U V ' ' ' ' ' Y V U U U V V X X X X ' ' ' ' ' ' ' ' ' APPLYING LIMITS ' ' DY DV DU DU U V DX DX DX DX DY DV DU U V DX DX DX D UV DV DU U V DX DV DX ' ' ' UV UV UV SOLVED EXAMPLES EXAMPLE 1: DIFFERENTIATE AX B CX D WITH RESPECT TO X SOLUTION: LET Y AX B CX D WE HAVE U AX B AND V = CX + D ' DU U A DX AND ' DV V C DX SUBSTITUTING IN DY DV DU U V DX DX DX I E , ' ' ' UV UV U V WE GET 'Y AX B C CX D A ACX BC ACX AD ACX BC AD WE CAN FIND THE DERIVATIVE AS SHOWN BELOW A TEXTBOOK OF ENGINEERING MATHEMATICS ALITER: ! " D D D AX B CX D AX B CX D CX D AX B DX DX DX AX B C CX D A ACX BC ACX AD ACX BC AC [ ] D AX B CX D ACX BC AD DX EXAMPLE 2: DIFFERENTIATE X X WITH RESPECT TO X SOLUTION: LET Y X X WE HAVE 1, = U X V X ' 2 , ' U X V X BY PRODUCT RULE ' ' ' Y UV U V ' 1 Y X X X X X X X X X X X ALITER D D D X X X X X X DX DX DX ! " X X X EXAMPLE 3: FIND THE DERIVATIVE OF (I) X LOG X (II) X X E (III) X X (IV) 1 LOG X X (V) SIN X X A TEXTBOOK OF ENGINEERING MATHEMATICS SOLUTION: (I) LOG LOG LOG D D D X X X X X X DX DX DX X X X X X X X E E E X (III) X X X X X X (IV) 1 LOG LOG LOG X X X X AX A A A X A A TEXTBOOK OF ENGINEERING MATHEMATICS DIFFERENTIATE THE FOLLOWING FUNCTION WITH RESPECT TO X 1 X X 2 2 LOG X X 3 4 LOG X X 4 X X E 5 X X 6 X X X X 7 X X X X 8 AX BX C PX Q 9 X XE 10 X X 11 TAN X X 12 1 LOG X X 13 X X 14 2LOG X X 3 ' 4LOG X X 4 XE X X 5 – 2 – X X 7 X X 8 P AX BX C AX B PX Q 9 1/ XE X X 10 X X A TEXTBOOK OF ENGINEERING MATHEMATICS SEC TAN X X X 12 1 / LOG X X X ! " 13 X X 15 QUOTIENT RULE IF F AND G ARE TWO DERIVABLE FUNCTIONS OF X, THEN ' – ' G X F X F X G X F X G G X ! " PROOF: LET F X F F X X G G X – – F X X F X F X X F X G X X G X ' ' ' – F X X G X G X X F X G X X G X ' ' ' – – F X X G X F X G X F X G X G X X F X G X X G X ' ' ' { – }– { – } G X F X X F X F X G X X G X G X X G X ' ' ! " ' X X F X X F X LT LT X G X X G X ' ' ' ' ' – – – X X F X X F X G X X G X G X LT F X LT X X ' ' ' ' ' ' ! " ' ' – ' F X G X F X F X G X G X G X ! " ' ' – ' G X F X F X G X F X G G X ! " A TEXTBOOK OF ENGINEERING MATHEMATICS ALTERNATIVE FORM: IF U AND V ARE TWO DERIVABLE FUNCTIONS OF X THEN – '– ' DU DV V U D U V U U V DX DX DX V V V PROOF: LET / Y U V THEN – – U U U Y Y Y V V V ' ' ' – U U V U V V Y V V V ' ' ' ' – – – UV V U UV U V V U U V V V V V V V ' ' ' ' ' ' – DU DV V U Y DX DX X V V V ' ' ' APPLYING LIMITS ON BOTH SIDES, WE GET 0 ( ) X X X X U V V LT U LT Y X X LT X LT V V V ' ' ' ' ' ' # ' ' ' ' ' ( 0) ( 0) DU DV V U DY DX DX X V DX V V # ' ' DU DV V U DX DX V # D U VU UV DX V V + + # SOLVED EXAMPLES EXAMPLE 1: DIFFERENTIATE AX B CX D WITH RESPECT TO X A TEXTBOOK OF ENGINEERING MATHEMATICS SOLUTION: LET AX B Y CX D THE TAKING U = AX + B, V = CX + D WE GET , + + DU DV U A V C DX DX BY QUOTIENT RULE DY VU UV DX V + + # ( ) ( ) ( ) # CX D A AX B C CX D ( ) ( ) ACX AD ACX BC AD BC CX D CS D # # # OR LET AX B Y CX D DIFFERENTIATING Y WITH RESPECT TO X, WE GET ( ) ( ) ( ) ( ) ( ) D D CX D AX B AX B CX D DY DX DX DX CX D # ( ) ( ) ( ) CS D A AX B C CX D # ( ) ( ) ACX AD ACX BC AD BC CX D CX D # # # EXAMPLE 2: FIND THE DERIVATIVES OF THE FOLLOWING FUNCTIONS WITH RESPECT TO X (A) X Y X (B) X X # (C) LOG X X (D) X X # SOLUTION: (A) X Y X A TEXTBOOK OF ENGINEERING MATHEMATICS DIFFERENTIATING WITH RESPECT TO X, WE GET ( 2) ( 4) ( 4) ( 2) ( 2) D D X X X X DY DX DX DX X # ( 2) X X X X # ( 2) ( 2) X X X X X X X # # # (B) WE HAVE X Y X # DIFFERENTIATING Y WITH RESPECT TO X, WE GET ( 1) ( 1) ( 1) ( 1) ( 1) D D X X X X DY DX DX DX X # # # # ( 1) X X X X X # # # ( 1) ( 1) # # # # # # X X X X X X X (C) LET LOG X Y X DIFFERENTIATING Y WITH RESPECT TO X, WE GET (LOG ) (LOG ) ( ) D D X X X X DY DX DX DX X # LOG X X X X X X # # (D) WE HAVE X Y X # A TEXTBOOK OF ENGINEERING MATHEMATICS ) # # # D D X X X X DY DX DX DX X ) X X X X X # # # ) XE F X X FIND ( ) F X + (C) FIND ( ) F T + IF T F T T # SOLUTION: (A) ( 1)( 2) ( ) ( 1)( 2) X X X X F X X X X X # # # # DIFFERENTIATING WITH RESPECT TO X, WE GET 2) ( 2) ( ) ( 2) D D X X X X X X X X DX DX F X X X # # # + 3) ( 2) X X X X X X X X # # # 6) ( 2) # # # # # # X X X X X X X X X X X X 12) X X X # A TEXTBOOK OF ENGINEERING MATHEMATICS ) XE F X X DIFFERENTIATING WITH RESPECT TO X, WE GET ) # + X X D D X E E X DX DX F X X ) # # X X X X X X E E E XE E X X ) X XE X (C) WE HAVE T F T T # DIFFERENTIATING WITH RESPECT TO X, WE GET ( ) ( 1) ( 1) ( 1) ( ) ( 1) D D T I T T T DX DX F X T # # # # + # ( 1)(2 ) ( 1)(2 ) ( 1) T T T T T # # # I FIND DY/DX FOR THE FOLLOWING: (A) X A X A # (A IS A CONSTANT) (B) X X # (C) X Y X (D) X X E Y E # (E) X Y X X II DIFFERENTIATE THE FOLLOWING FUNCTIONS WITH RESPECT TO X (A) X X X # (B) # SIN X SIN X (C) X X X # (D) X X X X # A TEXTBOOK OF ENGINEERING MATHEMATICS (E) X X (F) A X A X # (G) SIN X X III 1 IF T Y T # FIND DY DT 2 IF T F T T FIND ( ) F T + 3 IF I (A) ( ) A X A (B) ( 5) X # (C) ( 1) X X X # (D) ) X X E E # (E) 1) # X X X X II (A) ( 1) X X X (B) ) # COS X SIN X (C) ( 1) X X X # # (D) ) ( 1) X X X # # (E) ( 1) X (F) ( ) A A X # (G) # X COS X SIN X X III 1 ) DY DT T # 2 ) T F T T 3 AND U = X2 + DY U DU A TEXTBOOK OF ENGINEERING MATHEMATICS U X DU X DU ) (2 ) DY DY DU U X DX DU DX 6 ( 1) XU X X ALITER ( 1) 3( 1) ( 1) D D X X X DX DX # ! " 3( 1) 6 ( 1) X X X X EXAMPLE 2: IF Y X FIND DY/DX SOLUTION: LET U = X4 + WE HAVE Y U DY DX U AND DU U X X DX APPLYING CHAIN RULE DY DY DU DX DU DX X X U X OR DY D X DX DX X SOLUTION: (A) 3 LOG ) ] 3 LOG ) D D X X X DX DX # LOG ) X X X X (B) ( ) 4( ) ( ) # ! " D D X SIN X X SIN X X SIN X DX DX ) X SIN X COS X )( ) COS X X SIN X (C) 3/2 ( 1) ( 1) ( 1) D D X X X DX DX # ! " 3 ( 1) (2 ) 3 ( 1) 2 X X X X EXAMPLE 4: DIFFERENTIATE WITH RESPECT TO X (A) AX B E (B) LOG( 1) X X (C) X X # ! " (D) SIN X SOLUTION: (A) ( ) ( ) AX B AX B AX B D E E AX B A E DX (B) LOG( 1) ( 1) ( 1) D D X X X X DX DX X X ! " X X X (C) D D X X X X DX DX X X # # ! " # D D X X DX DX X X # ! " # D X DX X X X # # # ! " A TEXTBOOK OF ENGINEERING MATHEMATICS X X X X # # ! " X X X X X # # # ! " X # (D) ( ) ( ) D D SIN X COS X X DX DX COS X COS X X X CHAIN RULE CAN BE EXTENDED AS FOLLOWS: IF Y IS A FUNCTION OF U, U IS A FUNCTION OF V, V IS A FUNCTION OF W AND W IS A FUNCTION OF X, THEN DY DY DU DV DW DX DU DV DW DX EXAMPLE: FIND THE DERIVATIVE OF LOG( ! " X WITH RESPECT TO X SOLUTION: LET (LOG( 1) Y X ! " THEN LOG( 1) DY D X DX DX ! " 3 3 LOG LOG( 1) D X X DX # ! " ! " 3 LOG( 1) ( 1) ( 1) D X X DX X ! " A TEXTBOOK OF ENGINEERING MATHEMATICS 3 LOG( 1) ( 1) X X X ! " X E 5 X E# 6 5 8X E # 7 SIN AX B 8 1) X 9 1) X X # 10 ( ) AX BX C 11 1) X 15 ( 1) X 16 3) X X 17 ( )M AX B 18 AX BX C E 19 X X ! " 20 1 A/(AX + B) 2 1/(X LOG X) 3 2 (1 + LOG X) / X 4 3 E3X + 5 – E–X 6 (– 8) E5 – X A X # 12 LOG X X 13 SEC X X 14 3 (2X + 1) ½ 15 ( 1) X X # 16 10 ( 1)( 3) 3 X X X 17 ( )M MA AX B # 18 ) AX BX C AX B E 19 ) X X # # 21 DERIVATIVES OF INVERSE TRIGONOMETRICAL FUNCTION THE DERIVATIVES OF INVERSE TRIGONOMETRICAL FUNCTIONS ARE GIVEN BELOW (WITHOUT PROOF) 1 # # $ $ # D SIN X X DX X 2 # # # $ $ # D COS X X DX X 3 ( ) ; X R # D TAN X DX X 4 # # D COT X DX X