CHAPTER VII 



61. Successive Differentiation. When we are given y as a func- 

 tion of x and we find -j^, then -^- is also expressed as a function 



CLX ill 



of x, and therefore it is possible to differentiate again with respect 



d / f dti\ d^i/ 



to x. This process is represented symbolically by -T-(-T^) or -~ 



dx\(tX' dx 



In fact, with a few exceptions the process of differentiation can 

 be performed as many times as we please, and the result is repre- 



d n ii 

 sented symbolically by -5-^ where n gives the number of times 



the differentiation has been performed. 

 Thus if y = x n 



J. 



dx 



and -j = n(n- l)(n- 2) . . , (n-r+l)x n 



while taking n as a positive integer and putting r = n 



If n is a positive integer, x n can only be differentiated n times ; 

 but if n is a negative integer or a fraction, positive or negative, 

 there is no limit to the number of times x n can be differentiated. 

 For successive differentiation it is convenient, if possible, to find 



d n ii 



the general result of differentiating n times, i.e. -~, and then 



dx n 



any differential coefficient can be found by giving n the required 

 value in that general result. There are some functions for which 



99 



