Given a function $f(x,y)$, I would like to better understand the way the partial derivatives $ f_r(x(r,\theta),y(r,\theta)) $ and $f_\theta(x(r,\theta),y(r,\theta))$ are defined at the origin. In addition, at what conditions do these partial derivatives exist, and what exactly they represent.
To do so, I tried denoting $F(r,\theta) = f(x(r,\theta),y(r,\theta))$ and then to write the definition of the partial derivatives $F_r,F_\theta$ for $r=0$, but had no idea how to proceed. I am not even sure if the following expressions are correct,$$F_r(0,\theta) = \lim_{h\to 0 } \frac{F(0+h,\theta) - F(0,\theta) }{h }$$$$F_\theta(0,\theta) = \lim_{h\to 0 } \frac{F(0,\theta+h) - F(0,\theta) }{h }$$
Are those correct and correspond to the definition of the partial derivatives $F_r$ and $F_\theta$ at the origin?
In general, since $z=0$ correspond to only one value of r, but many of theta, what is the interpretation for $F_\theta(0,\theta) $ ?
As an example, if we consider $$ f(x,y)=\begin{cases} \frac{(x+y)^5}{(x^2+y^2)^2}, (x,y)\neq(0,0)\\ 0,\text{otherwise} \end{cases} $$ can we use polar coordinates and the definition of the partial derivatives w.r.t them in order to determine whether or not this function is differentiable at the origin?
My questions are still left unanswered, even after going over the following: Partial derivatives at the origin post
Thank you