DIFFERENTIAL COEFFICIENT OF AN EXPONENTIAL. 29 

 From Art. 19 the expressions 



Z V 



both tend to the limit unity. Hence we may assume 



g, s = 1 g 







where each of the quantities 7 and S has zero for its limit. 

 Hence 



vl+j> log, (l + i?) 



= n - from above ; 

 1 + 7 



At 



therefore the limit of - is n, and 

 z 







- = nx n 

 ax 



49. Differential coefficient of a". 

 Let y = a x , therefore 



y + Ay = a x+h = a x a h , 



therefore ~ = a 



Ace k 



Now, by Art. 20, the limit of 7 , when h is indefinitely 

 diminished is loge a; therefore 



Next let y = a" ; then 



