156 SECOND DIFFERENTIAL COEFFICIENT 



In this case we can verify our new rule, by comparing its 

 results with those previously found. In more complex 

 examples, such as 



we can find -/- only by the new method ; 

 dx 



putting u for x 5 ax s y + lx*y* y s , we have 





 therefore 



T- ) = ax* + 2bx*y 5y 4 ; 



*S* 



dy ox 



~r =-* ^r- ; - s 

 dx 5y 2 bx 2 y + ax* 



180. We shall now investigate the second differential 

 coefficient of an implicit function. 

 From the equation 



u or < (x, y) = 0, 

 (du\ 



we have deduced -- = , x (1): 



dx fdu\ 



\dy) 



d 2 y 

 it is required to find ~ . 



We observe that (I being a function of both x and y, 



\CLCCJ 



its differential coefficient with Respect to x must be found by 



Art. 172. If we put v for f-y-J, the required differential 



\dxj 



coefficient will be 



dv\ dy fdv\ 

 dy) dx \dxj ' 



Similarly, denoting ( -=- ) "by w, we have for its differential 

 \dy/ 



coefficient with respect to x, 



dw\ dy fdw\ 

 dyj dx \dx) ' 



