4 DIFFERENTIAL AND INTEGRAL CALCULUS. 



limit by putting d for A we obtain -j- = 2ait, and we have 



found the differential coefficient of the function OL of the 

 independent variable t. If then y = ax\ we shall have 



The first notion of a differential coefficient was that of 

 a velocity, time being in its nature an independent variable 

 which we cannot conceive except as uniformly increasing, 

 x the space described in a given time, then the speed, or 

 velocity, at the end of that time was called the fluxion of 

 x, and denoted by x. 



The general definition of a differential coefficient is as 

 follows; if?/ be any quantity depending on another magni- 

 tude x in such a way that the number representing y can 

 be expressed in terms of the number representing x, by 



such an equation as y= < (a 1 ), then the limit of ^ ' 



/i 



when h is indefinitely diminished, is the differential coefficient, 

 or first derived function, of y with respect to x, and it is 



denoted by either of the symbols ~ , </>' (x). 



This quantity will of course itself be a function of x, 

 and will have different values when different values are 

 given to x, thus <' (a) denotes the value of $' (x) when 

 x is put = a, </>' (0) its value when x is put = 0. If we 

 repeat the operation, the result is called the second diffe- 

 rential coefficient, or second derived function of x, and it 



is written either ~~ , or <" (x), and so on for any number 



GvMS 



of times. 



There is no harm in writing the equation -^ = </>' ( x ) 



in the form dy <' (x) dx, provided we bear in mind 

 that this means that the two members of the equation 

 &y = <' (x) Aie tend to have to each other a ratio of 

 equality when both are diminished indefinitely. When 



