24< EXAMPLE OF A DIFFERENTIAL COEFFICIENT. 



A?/ 



39. The limit of the fraction -~- , when A# is diminished 



Aa 



indefinitely, may be considered as affording a precise measure 

 of the rapidity with which the function increases when the 

 independent variable increases, for there remains no longer 



anything arbitrary 'in the expression. The limit ~- does not 



CiOC 



depend on the value assigned to A nor upon the form of 

 the curve at any finite distance from the point whose co-or- 

 dinates are araiid y; it depends only on the direction of the 

 curve at this point, that is to say, on the inclination of the 

 tangent line to the axis of x. 



40. As an example of the preceding, we will determine 

 the differential coefficient of V(a 2 ar"), and point out its 

 geometrical application. 



Let y V(a 2 #*), 

 then y + ky = V{- 2 - (a?4- ) 2 } ; 

 therefore A?/ = V (a 2 ( + ^) *} V (* #*)> 



_ 

 ~ VK - (*> 4- A)"} + V( a - ^ ' 



therefore - = -- 77-= ; -- ^-^j - 7T - i - -- . 

 Aa; V{ (a? 4 A) } + V( - ) 



The limit of this when h is made indefinitely small is 



x 



dy _ x 



therefore 



dx *j(a- x-) 



It will be seen that we have in the above example used an 

 algebraical artifice, namely, that of multiplying both numera- 

 tor and denominator of a fraction by V( a * (x+h) 2 }+*/(a* x 3 ], 



in order to obtain -^- in a form the limit of which can be 

 Ax 



