14 JOURNAL OF THE 



expression. If y =/{x) and we arbitrarily change x to 

 (x + //) = {x -\- A-r), then y will take a new value, desig- 

 nated by y + A7, where aj is the increase in y due to the 

 increase in x. 



Thus, {fy=/{x\ (i), 



;/ + Aj =/{x + /i) =/{x + A:r) (2), 



.V is here called the independent variable and y the de- 

 pendent one, since the value of y depends on that of .i-, which 

 we shall suppose to increase at will or independently of any 

 other variable in the formula. 



It is important to note here that although x is generally 

 increased so that ax is plus, yet the new value ofy ( = j -[- Ay) 

 niav be either greater or less than before. In the last case 

 Ay will be minus. Hence, if in any case. Ay is found ulti- 

 mately to be minus, we shall know how to interpret the 

 result. 



In equations (i) and (2), let x be first supposed to have a 

 fixed constant value, then j/ will have a corresponding con- 

 stant value. Subtracting (i) from (2) and dividing by a.t, 

 Ay /{x + ax) -/(..) 

 - = (3). 



A V AX 



We shall presently show that this expression generally 

 has a limit as ax approaches zero. From (2) above. Ay ap- 

 proaches zero indefinitely at the same time that ax does, so 



o Ay 



that (3) approaches indefinitely the form -, but the raho — 



O AX 



can never reach this form, for where Ay and ax are both 

 zero there is no ratio. We can however find the limif^ or 



Ay /(.r-f A.r)— /(.T) 

 the constant value to which the ratio — = 



AX Jx 



tends indefinitely zvithont ever being able to reach as a 

 ratio^ and this limit is known as the derivative^ derived 

 function or differential co-efficient of the function f(x) zvith 



