This seems to be essentially the same thing as "power series arithmetic" (first-order "dual arithmetic" is equivalent to arithmetic in the ring of formal power series modulo x^2, but you can make that x^n).
Encoding power series as matrices is sometimes convenient for theoretical analysis (or, as here, educational purposes), but it's not very efficient. The space and time complexities with matrices are O(n^2) and O(n^3), versus O(n) and O(n^2) (or even O(n log n) using FFT) using the straightforward polynomial representation (in which working with hundreds of thousands of derivatives is feasible). In fact some of my current research focuses on doing this efficiently with huge-precision numbers, and with transcendental functions involved.
Encoding power series as matrices is sometimes convenient for theoretical analysis (or, as here, educational purposes), but it's not very efficient. The space and time complexities with matrices are O(n^2) and O(n^3), versus O(n) and O(n^2) (or even O(n log n) using FFT) using the straightforward polynomial representation (in which working with hundreds of thousands of derivatives is feasible). In fact some of my current research focuses on doing this efficiently with huge-precision numbers, and with transcendental functions involved.