With complex functions, there are several paths that can be taken to approach a point $z_0$. We usually take the limit $$ f(z)' = \lim_{z \to 0} \frac{f(z_0+z) - f(z_0)}{z} $$ If I take an example, $w = z^3$,
$$ w' = \lim_{z \to 0} \frac{f(z_0+z) - f(z_0)}{z} $$
$= \lim_{z \to 0} \frac{(x+iy+z)^3 - (x+iy)^3}{z}$
And using Newton's binomial theorem,
$$ w' = 3(x+iy)^2 = 3z^2 $$
This is the correct derivative of z^3 but I don't exactly understand the mechanics at play here because if we check w' for uniqueness,
$\lim_{x \to 0} \lim_{y \to 1}$ will give us $w' = -3$ while $\lim_{x \to 1} \lim_{y \to 0}$ will give us $w' = 3$.
These are two distinct values, making the limit not unique, hence not analytic. But if the limit I used to obtain w' used z as it's variable, can I test the derivative with x and y? Or only with z?