48 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and so on, each differentiation adding -|TT to the angle, or 



d n . 



-7- (cosor) in exactly the same way = cosf x + J , or may 



be deduced from the last, for since 

 d . 



therefore 



d n , d n+ 



= cos 



d . 1 ^ 1 d 3 . . 1.2 



S (log ^ = i ' I 



and so on ; therefore 



The ?i h differential coefficients of sin (#-{-), cos(^c + a), 

 log(a;-f c) may be at once written down from the above, viz. 



d n . f . . / ^7T\ 



^-n sin (oa;) = a sin ( ax + I , 

 d JG \ ^ / 



--y-a cos (ax) = a cos ( ax + - 1 . 



rfiC \ 



and 



. o 1 cos2a; l 



Since sma; = - - -,cos;r = -, 



2 2 



3 sin# sin So; cos3a? + 3 coso; 



sin ic = - - - , cos 3 ic = - - -- , &c., 



the w tto differential coefficients of these and all like functions 

 can be at once obtained. 



The n th differential coefficient of (xu) is 



cTu d n ~ l u . d* . 



