592 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



1 a/ — TT 

 '2 



— ^ by differentiating the result in tlie last example. 



</4 d , 



(3) ■•• =^^--d^^''S^ 



rfa;* ' z 



IT 1 

 (-1)* -^ r by the fundamental formula 



/I 



X^ 



/— r 1 / 1 



= V-1 sVtt - 



1 / 1 



Hence all the results coincide. 

 Ex.8. To find ^i^S-^ 



d^zAogz_ d^Jogx^ 3 rf^ log z 

 —JJ '• rf,i ^2 dz^- 



Z , 1 3 -vZ — TT 



2 z"- ^ >}z 



_ -s/ — TT 



_ rf* log a; 



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 coefBcient of a product. 



Ex.4. To find -^;^. 



d" (iogzf _ 6?" log a: . log a; 



Here — -dlf^- 'd^" 



rf"loga: 11 rf"-Uoga; n{n — \) 1 rf""~^logr 

 '^'*'^*~"rfl'^ '''T f/x"-i 1.2 ■ ^ rfa:»-2~ 



,H«-l)(n-2) 1.2 ./"-Mogx _ ^^ 

 "*" 1.2.3 ' x' ■ t/a:"-' 



(/"—I log :(: 



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



