46 DIFFERENTIAL AND INTEGRAL CALCULUS. 



DIFFERENTIAL CALCULUS. 

 SUCCESSIVE DIFFERENTIATION. 



Having investigated methods of finding the differential 

 coefficients of any functions of #, we next consider rules 

 for the second, third, ... differential coefficients. We 



have denoted the differential coefficient of y by -7- ?/, i.e. 

 we may say that we denote the operation of differentiating 

 with respect to x by the symbol -j- , and a natural ex- 

 tension of the same notation will lead us to denote the 

 operation of differentiating 2, 3, ... n times by the symbols 



-T-j . [-=-] .... (-7- ). or the second, third, ... 7i tb diffe- 



dxj ' \dxj ' \dxj ' 



rential coefficients of y by -7-^ , ~][ , ... ^ . This nota- 

 tion has the great advantage that the indices denoting 

 the number of operations follow the laws of indices in 



,. . . . (d\ p '(d\ q fd\^ 



ordinary algebra; i.e. y = ( y, and any 



equation which holds in ordinary algebra between symbols 

 of quantities will hold also when membei's of the equation 

 are these symbols of operation. The most important for- 

 mula of successive differentiation is at once found in this 

 manner; we know that if u, v be two different functions 



of x, 



d , . du dv 



(UV} = V -7 h U -j- , 



dx ^ J dx dx 1 



or, we may write it, the operation -j- applied to the pro- 

 duct uv is equivalent to the sum of the operations Z> t and 

 D 2 , where J9, represents the operation of differentiating 

 u only, and Z> 2 that of differentiating v only. Hence the 

 operation 



