16 DIFFERENTIAL COEFFICIENT. NOTATION. 



Again, suppose < (x) = 

 therefore (* + A) = 



therefore (* + *)-(*) 



A (6 + x) (b + x + h) 



The limit of this when h = is 



which is therefore the differential coefficient of ^ - with 



b + x 



respect to x. 



25. We now give the notation which usually accompanies 

 the definition in Art. 24. 



Let < (x) = y, then <f> (x + K) <f> (x) is the difference of the 

 two values of the dependent variable y corresponding to the 

 two values, x and x + h, of the independent variable. This 

 difference may be conveniently denoted by the symbol Ay, 

 where A may be taken as an abbreviation of the word 

 difference. We have thus 



A*, = <(* + A) -<(*) 

 Agreeably with this notation, h may be denoted by Aa 1 , so that 



A.C h 



It may appear a superfluity of notation to use both h and 

 Aa; to denote the same thing, but in finding the limit of the 

 right-hand side we shall sometimes have to perform several 

 analytical transformations, and thus a single letter is more 

 convenient. On the left-hand side Ace is recommended by 

 considerations of symmetry. 



We say then, according to the definition in Art. 24, that 



AT/ 

 the limit of ~ , when Aa: is diminished indefinitely, is the 



faM? 



differential coefficient of y or <f> (x) with respect to a*. Tin's 

 limit is denoted by the symbol -j- . 



