588 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Section TI. — Differential Coefficients of Functions of x. 



Logarithmic Functions. 



18. Our &-st proposition must be the differential coefficients of log x. 

 Now we have already shewTi in Ex. 7 that, when n is a fraction, 



d" 3f' , ^ ,„ , , sin II TT , 



-^=(-1)" ,„ + l. -^Aogx 



omitting the constant. 



Hence, taking the differential of the - >ith order, we obtain 



„ / IN,, / 1 siu n TT d"" log X 



r" = (-l)" /n + 1 , 5 — 



TT dx~" 



d-" log X ^ (-!)-'• 71^ ^„ 



dx~" /n + \ sin « tt ' 



From the nature of the process, all the functions con-esponding with constants 

 of integration ai'e 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 



d^ ^ ' In 



therefore, taking the wth differential, we obtain 

 d" log X 



d^ =(-ir-/«.-". 



The last formula includes the former ; for 



r—^ TT 1_ 



sin n IT jn + 1 



dr^Xd^x , ,, . TT 

 = (-1)-". 



rfar"" sinwTT /« + l 



which is the result above. 



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

 proceed in the following manner. 



To find T^~ i'^'*<^*'"%- 



Since rf/- = rf;^Ti rf^l°g*- 



and that generally —r-y -—=- (-!)''• '-^ from the fundamental formula : 



