I am trying to bound the derivative of $$f(\boldsymbol{x},\boldsymbol{y})= \frac{\left(\sum_{i=1}^3x_i(x_i+y_i)\right)^2}{\sum_{i=1}^3(x_i+y_i)^2},$$ over the box $-a\leq x_i,y_i\leq a$.
To that aim I tried to find the limit $\lim_{\boldsymbol{x}\rightarrow -\boldsymbol{y}}f(\boldsymbol{x},\boldsymbol{y})$, but I failed in calculating it. I tried to use LHopital rule to calculate $\frac{\partial}{\partial x_i}f(\boldsymbol{x},\boldsymbol{y})$, but it is unclear to me how to do it in multivariable function. Do I take the derivative according to $x_i$ in all the iterations or should I take the mixed derivatives?
By change of variable $\boldsymbol{z}=\boldsymbol{x}+\boldsymbol{y}$, this amount to bound (on some bounded subset of $\mathbb R^6$) the differential of $$g(\boldsymbol{x},\boldsymbol{z})= \frac{\left(\sum_{i=1}^3x_iz_i\right)^2}{\sum_{i=1}^3z_i^2}.$$ But $$\frac{\partial g}{\partial z_1}((1,1,0),(2t,t,0))=-\frac6{25t}$$ is not bounded when $t\to0.$