CHAPTER V. 



SUCCESSIVE DIFFEEENTIATION. 



77. IN the preceding Chapters we have shewn how from 

 any given function of a variable another function may be 

 deduced, called the differential coefficient of the first. This 

 second function, by the same rules, has its differential co- 

 efficient, which is called the second differential coefficient of the 

 original function. 



Thus, if y = x n , we have ~ = nx n ~\ The differential co- 



efficient of nx"' 1 with respect to x is n (n 1 ) x n ~*, which is 

 therefore the second differential coefficient of y or x n with 

 respect to x. The second differential coefficient of y with 



respect to x is denoted by -~ 2 , which is to be considered as 



d dy 

 dx 



an abbreviation for 7 . 

 dx 



"What we said of -M in Art. 26, we now say of -j*. 

 dx dx s 



that it is to be looked upon as a whole symbol, not admitting 

 of decomposition into a numerator d"*y and a denominator dx' 2 . 



d*y 

 As -JTJ will be generally a function of x it will have its 



CLOC 



differential coefficient with respect to x. This is called the 

 third differential coefficient of y with respect to a?, and is 



d^y 



J3 . dx* 



denoted by -3% , as an abbreviation for ' . 



This process and notation may be carried on to any extent. 



