Suppose I have a function $f(\boldsymbol{x})$ defined on $\mathcal{X}$. I would like to compute this $$ \nabla_{\boldsymbol{x},\, \boldsymbol{y}} \int_{\mathcal{X}} f(\boldsymbol{x}) d\boldsymbol{x} $$ where $\boldsymbol{y}$ is some other set of variables. Basically I am taking the gradient not only with respect to $\boldsymbol{x}$ but also with respect to $\boldsymbol{y}$. I have essentially defined a new variable $\boldsymbol{\xi} : =(\boldsymbol{x}, \boldsymbol{y})$ and then I am taking the gradient $\nabla_{\boldsymbol{\xi}}$.
My Working
I think that $\nabla_{\boldsymbol{x}, \boldsymbol{y}} f(\boldsymbol{x}) = (\nabla_{\boldsymbol{x}} f(\boldsymbol{x}), \nabla_{\boldsymbol{y}} f(\boldsymbol{y}))$ and so $$ \nabla_{\boldsymbol{x},\, \boldsymbol{y}} \int_{\mathcal{X}} f(\boldsymbol{x}) d\boldsymbol{x} = \int_{\mathcal{X}} (\nabla_{\boldsymbol{x}} f(\boldsymbol{x}), \nabla_{\boldsymbol{y}} f(\boldsymbol{y})) d\boldsymbol{x} = \int_{\mathcal{X}} \nabla_{\boldsymbol{x}}f(\boldsymbol{x}) d\boldsymbol{x} + \int_{\mathcal{X}}\nabla_{\boldsymbol{y}} f(\boldsymbol{x}) d\boldsymbol{x} = \int_{\mathcal{X}} \nabla_{\boldsymbol{x}}f(\boldsymbol{x}) d\boldsymbol{x} + 0 = \int_{\mathcal{X}} \nabla_{\boldsymbol{x}}f(\boldsymbol{x}) d\boldsymbol{x} $$ Does this make any sense?