66 EXAMPLES OF SUCCESSIVE DIFFERENTIATION. 



The 71 th differential coefficient of . - 5 with respect to x 



a* + x 



is sometimes obtained thus : 



_J_ = _ L_ f * * 1. 



a* + x* 2aV(-l) laj-aV(-l) + aV(-l)J' 



therefore 



d* / 1 \ = (-l) n l!L 

 dx n {cf+ef} 2a V(- 1) 



Now assume x = r cos 0, a = r sin 0, so that 



3 = a 3 + a? and tan = - . 

 x 



Then {a + a V(- 1) j n+1 = ^ n { cos e + -J(~ 1) sin 



= r n+l (cos (i + 1) 6 + V(~ 1) sin 

 by De Moivre's Theorem. 

 Hence 



and we ; obtain the same result as before for the proposed ?i th 

 differential coefficient. 



j 



tt 



. 



Art ' 80 - 



Hence, by means of the preceding Example, shew that 



(-l)"[cos(n-f 1)0 



d n f x \ 

 dx n \f? + a?) ~ 



(a 8 -fa; 2 )' 



We may also proceed in the second manner indicated for 

 the preceding Example, starting with 





a* + a 2 2 [x + a V(- 1) 



*-aV(-l)r 



