I'm trying to analytically find the gradient the log a multivariate Gaussian density, which is given by
$$ f(x_{1}, \dots, x_{p}) = f(\vec{x}) = \frac{1}{(2\pi)^{p/2}|\Sigma|^{1/2}} \exp \Bigg(-\frac{1}{2}(\vec{x} - \vec{\mu})^{T}\Sigma^{-1}(\vec{x} - \vec{\mu})\Bigg) \sim \mathcal{N}(\vec{\mu}, \Sigma) $$
where $\vec{x}, \vec{\mu} \in \mathbb{R}^{p}$, and $\Sigma \in \mathbb{R}^{p \times p}$.
I want to find the analytical expression for $\nabla f$, where $\nabla$ is the gradient operator defined as
$$ \nabla f(\vec{x}) = \Bigg[ \frac{\partial f(\vec{x})}{\partial x_{1}}, \dots, \frac{\partial f(\vec{x})}{\partial x_{p}} \Bigg] $$
My vector calculus is very rusty and not sure how to take partial derivatives with respect to $x_{1}, \dots, x_{p}$.
Any help is much appreciated!