DIFFERENTIAL COEFFICIENT OF A SUM. 17 



CM ?/ 



26. The symbol - we consider as a whole, and we do 



not assign a separate meaning to dy and dx. As, however, 



\ti 



^- is a real fraction in which Ay and Aa; have definite 



meanings, the student will very possibly suspect that some 

 meanings may be given to dy and dx which will enable him 



to regard -j- as a fraction. This suspicion will probably be 

 strengthened as he proceeds in the subject and finds that in 

 many cases -,'- possesses the properties of an algebraical 



fraction. We remark that there are indeed methods of 

 treating the Differential Calculus in which meanings are 

 given to dy and dx, and we shall recur to them hereafter 



(see Chap, xxvii.), but at present we define the symbol j- 



as above, and only leave to the reader the task of examining 

 whether we are consistent with ourselves in the inferences 

 we proceed to draw and express by means of our definitions 

 and symbols. 



The following notation is also frequently used. If <j> (x} 

 denote any function of x, then <j>'(x) denotes the differential 

 coefficient of <f> (x) with respect to x. 



The operation of finding the differential coefficient of 

 a function is called "differentiating" that function. 



27. Differential coefficient of a sum of Functions. 



Let y and z denote two functions of x, and u their sum. 

 Suppose that y, z, u', denote the values these functions 

 assume when x is chaned into x + h. Then 



u' = y' + z, 



therefore u u = y' y -f z 3 ; 



that is Aw = Ay + A. 



Divide by h or A#, then 



Aw _ A.y Az 

 A# A# A# ' 

 T. D. C. 



