Yes, > 1 for any prime; everything else calculated via the product rule does indeed have a surprising amount to do with differentiation! If you take the usua

Yes, > 1 for any prime; everything else calculated via the product rule does indeed have a surprising amount to do with differentiation! If you take the usual polynomial functions in one variable (lets say x is the variable and all our things are complex numbers) then these can be factored: e.g. x^2 + 3x + 2 = (x+1)(x+2). They form a (so called) unique factorization domain, which essentially means that factorization into "primes" works exactly the same as it does for integers. In the example above (x+1) and (x+2) are examples of prime factors which can't be factored any further. If you take the definition "1 for any prime; everything else calculated via the product rule" and apply it to this system where our "numbers" are polynomials and our "primes" are the polynomials we can't factor any further you get a definition of an "arithmetic derivative" for polynomials. The fun fact then is that this arithmetic derivative we just defined is exactly the same as the usual definition of the derivative from calculus: D[(x+1)(x+2)] = (x+1)D[(x+2)] + (x+2)D[(x+1)] = (x+1) + (x+2) = 2x+3 whereas d/dx (x^2 + 3x + 2) = 2x + 3