592 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



,,,, d"^ \o^x d di , 



(2) — = —— . —-^ log X 



^ ' ^^% dx dx^ ^ 



= ^ ■ — I — by differentiating" the result in the last example. 



rf^ 1 



dx^ X 







= (-1)^ 



' 2 



/I 



1 



3 







= ^/-l 



^Vtt 



1 



xi 



? all the results coincide. 



- 1 



• 



i:x. 3. 



To find 



d ^ X log X 



dx^ 



-• 





by the fundamental formula 



rf'a;.logar_ rf^log.r 3 d^ log z 

 ""' ' "^"2 dx^ 



dx- dx^ 



X . 1 3 \/ — TT 



= — ^_7r -T-- 2—7 



■^ X^ ^ ^ X 



V- 



TT 



_ c?* log X 

 ^ dx'- ' 



If we desire to obtain the differential coefficients of powers or other functions of 

 log X, we have, in general, no other way of proceeding than to adopt the series 

 for the differential coefficient of a product. 



Ex. 4. To find 

 Here 



dx"- 



(^" (loga;)^ _ c?" log « . log « 

 d^' ~ rf^» 



_ c?"logx n d'^-^\ogx n{n — V) 1 rf""~^loga: 



~ "^ ^ rf.r" "^ ~x dx"-^ 1.2 ■ ^ ~^^-2~" 



n(>^-l)(n-2) 1.2 </"-^ log a: 

 ^ 1.2.3 ■ :*;3 • dx^^ 



d^ — ^ loff X 



Let 1 + P = Q„ , and write Q^i &c. for the functions corresponding to ^f 



m5 it 



