EXAMPLES OF SUCCESSIVE DIFFERENTIATION. 65 



lence 



d z y 



-j4 

 ax 



2 y . y dy 



cos^sm^-/ 



a a a ax 



I . Zy dy 



= - - sm -/- 



a a ax 



Shew that -^. = -= cos [ + 2 . ) cos 3 - , 

 ax a \a 2 / a 



and generally that -=-*. = - -^ cos \ -*- + (n 1) [ cos" - . 

 dx n a (a 2) a 



Now tan" 1 - ' = - tan" 1 - = --0 suppose ; 

 a 2 x 2 



i ,\ ir\ fny WTrN . , ... 



+ (n 1) -V = sin I + J = sin (nir nV) 



thus cos \ 



therefore 



19. Since 



; and cos n ^ 

 \n 





sin 



(Ztan 1 - 



a a 



\rt 



Hence, shew that 



where 



T. D. c. 



tan ^ = - 



