58 SUCCESSIVE DIFFERENTIATION. 



The successive differential coefficients of a function are 

 often conveniently denoted by accents on the function. 

 Thus, if <(#) be any function of x, then <' (x), (j>"(x), <f>'"(&), 

 (^(x}, ...... denote the first, second, third, fourth, ...... 



differential coefficients of <j>(x) with respect to x. 



78. In some cases the n ih differential coefficient of a 

 function admits of a simple algebraical expression. For 

 example, suppose 



y sin x ; 



therefore -p = cos x = sin ( x + ) , 



diX \ 2/ 



,. d sin ( x + } 

 tf'V \ 2/ 



-5 

 dx 



e + - 



so 



, d*y . / nTr\ 



and generally ^ = sm f + yj 



So also, if y = sin ax, 



d n 



In like manner, if 



y = cos ar, 



, 

 and 



- f <Z n v / , WTT\ 



if ._y = cosax, -^ = a" cos f ax + -z- J 



