Differentiation and its Uses in 
                            Business Problems      8 
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
          The  objectives  of  this  unit  is  to  equip  the  learners  with 
          differentiation and to realize its importance in the field of business. 
          The  unit  surveys  derivative  of  a  function,  derivative  of  a 
          multivariate functions, optimization of lagrangian multipliers and 
          Cobb-Douglas production function etc. Ample examples have been 
          given  in  the  lesson  to  demonstrate  the  applications  of 
          differentiation  in  practical  business  contexts.  The  recognition  of 
                     School of Business 
                     differentiation  in  decision  making  is  extremely  important  in  the 
                     filed of business. 
                      
                      
                      
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                     Unit-8                      Page-164 
                                                                          Bangladesh Open University 
                       Lesson-1: Differentiation 
                       After studying this lesson, you should be able to: 
                           
  Explain the nature of differentiation; 
                           
  State the nature of the derivative of a function; 
                           
  State some standard formula for differentiation; 
                           
  Apply the formula of differentiation to solve business problems. 
                       Introduction 
                       Calculus is the most important ramification of mathematics. The present 
                       and potential managers of the contemporary world make extensive uses 
                       of this mathematical technique for making pregnant decisions. Calculus 
                       is inevitably indispensable to measure the degree of changes relating to 
                       different  managerial  issues.  Calculus  makes  it  possible  for  the 
                       enthusiastic and ambitious executives to determine the relationship of 
                       different  variables  on  sound  footings.  Calculus  in  concerned  with 
                       dynamic situations, such as how fast production levels are increasing, or 
                       how rapidly interest is accruing. 
                       The  term  calculus  is  primarily  related  to  arithmetic  or  probability 
                       concept.  Mathematics  resolved  calculus  into  two  parts  -  differential 
                       calculus and integral calculus. Calculus mainly deals with the rate of 
                       changes  in  a  dependent  variable  with  respect  to  the  corresponding         Differential calculus 
                                                                                                          is concerned with the 
                       change in independent variables. Differential calculus is concerned with 
                                                                                                          average rate of 
                       the average rate of changes, whereas Integral calculus, by its very nature, 
                                                                                                          changes. 
                       considers the total rate of changes in variables. 
                       Differentiation 
                       Differentiation is one of the most important operations in calculus. Its 
                       theory  solely  depends  on  the  concepts  of  limit  and  continuity  of 
                       functions.  This  operation  assumes  a  small  change  in  the  value  of 
                                                                                                          The techniques of 
                       dependent  variable  for  small  change  in  the  value  of  independent 
                                                                                                          differentiation of a 
                       variable. In fact, the techniques of differentiation of a function deal with       function deal with the 
                       the  rate  at  which  the  dependent  variable  changes  with  respect  to  the    rate at which the 
                       independent  variable.  This  rate  of  change  is  measured  by  a  quantity      dependent variable 
                                                                                                          changes with respect 
                       known  as  derivative  or  differential  co-efficient  of  the  function. 
                                                                                                          to the independent 
                       Differentiation  is  the  process  of  finding  out  the  derivatives  of  a       variable. 
                       continuous function i.e., it is the process of finding the differential co-
                       efficient of a function. 
                       Derivative of a Function 
                       The  derivative  of  a  function  is  its  instantaneous  rate  of  change. 
                       Derivative is the small changes in the dependent variable with respect to 
                       a very small change in independent variable. 
                                                      dy
                       Let y = f (x), derivative i.e.      means rate of change in variable y with 
                                                      dx
                       respect to change in variable x. 
                       Business Mathematics                                                 Page-165 
                                                            School of Business 
                                                            The  derivative  has  many  applications,  and  is  extremely  useful  in 
                                                            optimization- that is, in making quantities as large (for example profit) 
                                                            or as small (for example, average cost) as possible. 
                                                             
                                                            Some Standard Formula for Differentiation 
                                                            Following are the some standard formula of derivatives by means of 
                                                            which we can easily find the derivatives of algebraic, logarithmic and 
                                                            exponential functions. These are : 
                                                                 dc
                                                            1.       = 0, where C is a constant. 
                                                                 dx
                                                                    n
                                                                dx        d                     
                                                                                 n         n-1
                                                            2.        =       [x ] = n. x
                                                                 dx      dx
                                                                  d                 d            
                                                                          ( )             ( )
                                                            3.       a.f x =a          [f x ]
                                                                 dx                dx
                                                                       1             −(n+1)
                                                                d      −          1
                                                            4.     x n  = −       .x   n    
                                                               dx               n
                                                                         
                                                                    x
                                                                de        d               
                                                                             ( x )     x
                                                            5.        =       e    =e  
                                                                 dx      dx
                                                                  d    g(x)       g(x)  d
                                                                                              ( )
                                                            6.       [e     ]=e       .    [g x ] 
                                                                 dx                    dx
                                                                                                        dy     dy     du
                                                            7.  If y = f(u) and U = g(x) then               =      ×       
                                                                                                        dx     du     dx
                                                                 d
                                                                    ( x )      x           
                                                            8.       a    =a .log a
                                                                                      e
                                                                dx
                                                                  [ ( )       ( )]       [ ( )]       [ ( )]
                                                                d f x ± g x            d f x        d g x
                                                            9.                      =            ±               
                                                                        dx                dx           dx
                                                                  d                 d              1
                                                                      (        )       (       )
                                                            10.        log x =          ln x =          
                                                                           e
                                                                  dx               dx              x
                                                                                  n           dy                     d[f (x)]
                                                                                                               n
                                                            11.  If Y = [f(x)]    then,           = n [f(x)] –1 .               
                                                                                              dx                        dx
                                                                   d             1
                                                            12.        log x =     log e 
                                                                           a              a 
                                                                   dx            x
                                                                     [ ( ) ( )]                [ ( )]             [ ( )]
                                                                   d f x .g x                d g x              d f x
                                                            13.                     = f (x)            + g(x)               
                                                                         dx                     dx                  dx
                                                            Unit-8                                                                                Page-166