586 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



dU^- , ... 1 /I- ,1 



By formula (3) ^t^ ^{-Xf 



ax - /f .r 



This example will serve us to verify the very singular and unexpected result 

 which we obtained in Ex. 4. 



d^x^ __ dd} x^ _ d aZ-I 

 F«r -J^ ~J7J^^-dx ■ -27nr ^'S- (by Ex. 4.) 



= ~cr~r — • — which coincides with the result above. 



2 V TT X 



The verification here obtained is stronger than at first sight may appear, in- 

 asmuch as the result of Ex, 6 was obtained without having recourse to the in- 

 troduction of an arbitrary constant, whereas that in Ex. 4 depended entirely on 

 the arbitrary constant. 



d" x" 



Ex. 7. Find the value of -r— - when n is a fraction. 



««" 



By the second formula, 



dx"' fT ' sin(w — /x)7r 



sin w TT , X 



jn- 



— a 



— ^ 



= (-!)'• /« + ! . log. — . 



dh^ 

 Ex. 8. Find the value of 



dx^ 

 By the second formula, 



d^ X , ^,i /z sin . TT 1 



dx^ ^ /f srn^TT 



16. This example appears to have been the first to induce men to think on 

 the subject. 



EuLER, in the Petersburgh Commentaries, vol. v. for 1730, gives the follow- 

 ing result as the basis of general differentiation. 



fdx( — \osx^' 



d" z' = 2*-" dz"" -^ ; ^ ^ ( 



fdx{-\ogxy 



e — n 



and obtains fi*om it, by putting e=l, «=y» ^*^~n/^t~ • ' where A is a constant. 

 If A = 00 , this result coincides with that which we have deduced. 



