5gg PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



rfl^ . , ,s 1 /¥ T 



By formula (3) ~^x = ( - Ij ■ :^ " 



V-1 1 



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

 which we obtained in Ex. 4. 



d^x^ ,/d-x- d \/-l 



F«^ ~Ixi~771^i-dcc--YJ^^-"^ (by Ex. 4.) 



= "^Lr^ — . — which coincides with the result above. 



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. 



of" x" 



Ex. 7. Find the value of -^— ^ when w is a fraction. 

 By the second formula, 



di'-z" _/_2y, /« + ! sinnTT ,n—^ _„"— ^ ) 



dxi' iT ' Bm(n-fjL)7r 



/ i\,. / r sinraTT x 



^ ' ' rjr ^ a' 



Ex. 8. Find the value of j- 



dx " 



By the second formula. 



dx^ /f sinl-T 



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. 



fdxi — losxy 



^n »e _ j,e— n fj^n _J. v a — i _ - 



/(/«(- log «)'-" 



and obtains from it, by putting e=l, w^-g' '^^^"n' -j- • > '"^here A is a constant. 

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





