Given an expression, for example, $f(x,y)=\frac{x}{(x-1)(1-y)(xy+x-1)}$
How can we find the general expression involving $m$ and $n$ for $\lim_{x\to 0, y\to 0}{\frac{\partial^{n+m}{f}}{\partial{x}^{m}\partial{y}^{n}}}$
Given an expression, for example, $f(x,y)=\frac{x}{(x-1)(1-y)(xy+x-1)}$
How can we find the general expression involving $m$ and $n$ for $\lim_{x\to 0, y\to 0}{\frac{\partial^{n+m}{f}}{\partial{x}^{m}\partial{y}^{n}}}$
Suppose $f = \sum_{i,j \geq 0} c_{ij} x^i y^j$. The limit you're looking for is $m!n! c_{mn}$. Your function is better written as $f(x,y) = \frac{x}{(1-x)(1-y)(1-x-xy)},$ where the most interesting factor is $\frac{1}{1-x-xy} = \sum_{t \geq 0} x^t(1 + y)^t = \sum_{t,s \geq 0} \binom{t}{s} x^t y^s.$ Multiplying by a variable $1/(1-z)$ is like taking "running sums" across $z$. So $\frac{x}{(1-x)(1-y)(1-x-xy)} = \sum_{t,s \geq 0} x^{t+1} y^s \left[ \sum_{T \leq t, S \leq s} \binom{T}{S} \right]. $ So the value you're looking for is $m!n!\sum_{T \leq m-1,S \leq n} \binom{T}{S}.$