Calculations with dual numbers for computing derivatives.
If I is the identity matrix and J is the matrix with 1's above the main diagonal and 0's everywhere else, then a
dual number has the form
a = a0 I + a1 J + a2 J2 + ...
For sufficiently smooth functions
f(xI + J) = f(x) I + f'(x) J + f"(x)/2! J2 + ...
This allows us compute derivatives without taking limits of difference quotients.
(xI + J)3 = x3 I + 3 x2 J + 3 x J2 + J3.
We can read off the derivatives of f(x) = x3: f'(x) = 3 x2 and f"(x)/2! = 3 x, so f"(x) = 6x and the third derivative is 3!.