In a polynomial ring $R[x]$, the “formal derivative” seems to have two formulas, the classic one:
$$ f’(x) = a_1 + 2 \cdot a_2 \cdot x + 3 \cdot a_3 \cdot x^2 + \cdots $$
(if $f(x) = a_0 + a_1x + a_2x^2 + \cdots$).
But it also seems like this $g(x,x)$ works as well, if $g$ is defined as:
$$ g(x, y) := \frac{f(x) - f(y)}{x - y}. $$
but to me $g(x, x)$ is trivially 0, so I’m confused here :d.
$g(x, x)$ is not trivially 0. $$g(x, x) = \frac{f(x)-f(x)}{x-x} = \frac{0}{0},$$ which is not defined!