let e^2 = 0. the dual numbers are D = {x + ye | x,y in R}. let f(z) be some polynomial. Then by binomial theorem f(z + e) - f(z) = z^n + n*z^(n-1)*e + O(e^2

let e^2 = 0. the dual numbers are D = {x + ye | x,y in R}. let f(z) be some polynomial. Then by binomial theorem f(z + e) - f(z) = z^n + n*z^(n-1)*e + O(e^2) - z^n = f'(z)*e. And so df = df/dz * e