A Scaled Squared Distance to a Point is of Class $C^2$, with Gradient and Hessian

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $a=(a_1,\dots,a_n)\in\mathbb{R}^n$, and let $c$ be an element of the real numbers.

Write $d_E$ for the Euclidean distance, $x-a$ for the difference of points of $\mathbb{R}^n$, $\lambda z$ for the scalar multiple of $z\in\mathbb{R}^n$, $I_n$ for the identity matrix of size $n$, and $\mu M$ for the scalar multiple of a real matrix.

Let $q:U\to\mathbb{R}$ be the function whose value at $x\in U$ is

$$q(x)=c\,d_E(x,a)^2 .$$

Then the following hold.

1. (First partial derivatives) For every $x=(x_1,\dots,x_n)\in U$ and every $j\in\{1,\dots,n\}$ the partial derivative of $q$ with respect to the $j$th variable exists at $x$, and

$$\frac{\partial q}{\partial x_j}(x)=2c\,(x_j-a_j).$$

2. (Regularity) $q$ is of class $C^2$ on $U$.

3. (Gradient and Hessian) For every $x\in U$ the gradient and the Hessian matrix of $q$ at $x$ are

$$Dq(x)=(2c)(x-a),\qquad D^2q(x)=(2c)I_n .$$