100 DIFFERENTIATION OF A FUNCTION OF TWO VARIABLES. 

 Now, by Art. 92, 



/" (x + h) =/" to + hf (a) + -- /'"* (x + 6h], 



therefore 



Ay = h n /" (x) + A B+a (|/" +2 (a; + 0A) + ^ (a? 



=h n y n+i (x} + h^ 1 (x) ..................... (2). 



Equation (2) shews us that, granting the truth of (1), we 

 can deduce for A"" 1 " 1 ^ a value of the same form as that we 

 assumed for A"y. But Art. 127 gives for A*y an expression 

 of the assumed form ; hence A 8 y has the same form, and so 

 also has A 4 #, and generally A"y. 



From equation (1), by dividing both sides by h n and then 

 diminishing h indefinitely, we have 



the limit of ( ||^ =/"(*); 



A"?/ . d n v 



that is, the limit of , . . is -~= . 

 (Ax)" dx n 



129. Hitherto we have only considered functions of one 

 independent variable; that is, we have supposed in the equa- 

 tion y =f(x), although quantities denoted by such symbols 

 as a, b, ... might occur in f(x], yet they were not susceptible 

 of any change. Suppose now we have the equation 

 u = x 2 + xy + 7/ 2 , 



and let y denote some constant quantity and x a variable, 

 we have 



du 



From the same equation, if a; be a constant quantity and y 

 a variable, we obtain 



du 



Of course we cannot simultaneously consider x both con- 

 stant and variable ; but there will be no inconsistency if on 

 one occasion and for one purpose we consider x constant, 

 and on another occasion and for another purpose we consider 

 it variable. 



