CHAPTER II. 



DEFINITION OF A DIFFERENTIAL COEFFICIENT, 



DIFFERENTIAL COEFFICIENT OF A SUM, PRODUCT, AND 



QUOTIENT. 



24. WE shall now lay down the fundamental definition 

 of the Differential Calculus, and deduce from it various 

 inferences. 



DEFINITION. Let < (x) denote any function of x, and 

 <f> (x -}- /*) the same function of x + h ; then the limiting 



value of -- j , when h is made indefinitely small, 



is called the differential coefficient of <f) (x} with respect to x. 



This definition assumes that the above fraction really lias 



a limit. Strictly speaking, we should use an enunciation of 



this form " If - j have a limit when 7^ is made 



h 



indefinitely small, that limit is called the differential coefficient 

 of $ (x) with respect to x" We shall shew, however, that 

 the limit does exist in functions of every kind, by examining 

 them in detail in this and the following two Chapters. We 

 give two examples for the purpose of illustrating the defini- 

 tion. 



Suppose < (x) = x* ; 



therefore <f>(x + h} = (x 



there f ore *( + *)-*() _ 



k It 



Ixh 4- 7? 2 



T" " 



and the limit of 2x + h when h = 0, is 2x ; therefore 2x is the 

 differential coefficient of or with respect to x. 



