588 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Section II. — Differential Coefficients of Functions of x. 



Logarithmic Functions. 



18. Our first proposition must be the differential coefficients of log- jc. 

 Now we have already shewn in Ex. 7 that, when /^ is a fraction, 



omitting the constant. 



Hence, taking the differential of the — y^th order, we obtain 



, - , , ^- sin n TT rf""" log x 



TT 



d^'logx _ (-1)-" 



TT 



sinnTT 



From the nature of the process, all the functions corresponding with constants 

 of integration are necessarily omitted, and these may, as we shall see presently, 

 embrace the most important part of the result. 



Again, we saw in Art. 8. that 



dx-^ 



I 



n 



therefore, taking the nth differential, we obtain 

 The last formula includes the former ; for 



, TT 1 



/-n=- 



d~"\ogx 



sinmr 



/n + l 



=(- 



-ir 



n '^ 



• 



/^^l 



dx"" " ' smnTT 



which is the result above. 



19. To obtain a more general form in each case, if there be one, we must 

 proceed in the following manner. 



To find ^^ — generally. 



dx"" 



Since 



^"losrx rf«+i d-^ , 

 dx"" rfa;" + i dr-^ * 





d'^ X 



and that generally —^ 



/-I 



(-1X 



//•-I 



,»r— 1 



from the fundamental formula ; 



