I have a function $f \colon \mathbb{R}^{d \times d} \times \mathbb{R}^d \to \mathbb{R}$ defined by $$ f(A, c) = (z - c)^T A (z - c) - 1 $$ where $A \in \mathbb{R}^{d \times d}$ is symmetric positive definite, $c \in \mathbb{R}^d$, and $z_i \in \mathbb{R}^d$ is a given constant. This function is used in defining an optimization problem, and so it is of interest to find wether or not this function is in $C^1$ and $C^2$. I know that for simpler functions, i.e. 1-dimensional functions $g \colon \mathbb{R} \to \mathbb{R}$, this simply involves showing that the function is once and twice differentiable. However, this problem has me confused. How would I even go about computing the gradient of a function like this?
All help is appreciated. If anything is unclear or further clarification is needed, please let me know.