294 



Let 



then 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



x'" (x + 1)" 



+ &C. 



Ex.9. I'o find A"^ ain a z. 



d 



A" sin ax= ^^^ (e'*^ - 1)" (e"-^'^^ -.-"^'^-1) 



} 



= i(e -e fe 



-{-rr{e 



-V3i -5V-I — 5-V-l + aa:V-l 



) 



-(co8 2X + l»i7r-V'^l sin2X + l«'T) {coaax + -^ -\/-l sin a.r + — j | 

 = (2 -v/^D""^ sin s [ cos ( aa;+ y j + a/^ sin (oa;+ ^j 



— cos \2X + lnv + ax+ -^\ +1/ -l9,m\2\ + ln'jr + ax + ^\ \ 



na 2X+1 



= 2" y^" - 1 sin" ^ sin (a X + ^" + =^^ » TT) (sin 2 \ + 1 :^ &c. 

 =2"(co8»y'7r— a/— 1 sin n r if) sin"^ sin (ox+ -=- + ^^ — ^~^^'^) ' 



r and \ being any integers. 



Ex. 10. To find a" e" ■" sin a «. 



dx 



ax=—j=. (e"-^ -If {e"'^+'""J -^ _^m.r-o..^ v-i) 

 2 V — 1 



2V-I 



-7n X ____ 



= y — :^ {(e cosa + e V— Isin^i— 1) (coB«^+ v — 1 sin a:c) 

 2 V — 1 



