If you are going to use infinitesimals, though, it requires some additional doing for notation. The standard notation for higher-order derivatives (and partial derivatives) needs to be modified in order for them to work (but they do work great once you do this).
Instead of the second derivative being "d^2y/dx^2" it is "(d^2y/dx^2) - (dy/dx)(d^2x/dx^2)" and the differentials can be manipulated just like any other entity. Additionally, you can infer this notation by simply applying the quotient rule to the first derivative (which is a quotient of infinitesimals).
See more:
"Extending the Algebraic Manipulability of Differentials" ( 10.48550/arXiv.1801.09553 )
"Total and Partial Differentials as Algebraically Manipulable Entities" (
10.48550/arXiv.2210.07958 )
Instead of the second derivative being "d^2y/dx^2" it is "(d^2y/dx^2) - (dy/dx)(d^2x/dx^2)" and the differentials can be manipulated just like any other entity. Additionally, you can infer this notation by simply applying the quotient rule to the first derivative (which is a quotient of infinitesimals).
See more:
"Extending the Algebraic Manipulability of Differentials" ( 10.48550/arXiv.1801.09553 )
"Total and Partial Differentials as Algebraically Manipulable Entities" ( 10.48550/arXiv.2210.07958 )