I don't like this use of dual numbers notation for differentiation because division by a dual number is not defined. For example, how would one calculate ε^2/ε?

I don't like this use of dual numbers notation for differentiation because division by a dual number is not defined. For example, how would one calculate ε^2/ε? If ε is a matrix [0,1;0,0] then it doesn't have inverse and thus the expression could not be evaluated. On the other hand, little-o notation [1] was invented for exactly this purpose. It is easy to evaluate derivatives using it, for example (x+ε)^3 = x^3+3x^2ε+o(ε), and so ((x+ε)^3 - x^3)/ε = 3*x^2 + o(1), and o(1) tends to 0 for ε tends to 0. [1] https://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation https://en.wikipedia.org/wiki/Big_O_notation#Little-o_notati...