Let $A_k$ be the $n\times n$ matrix defined by $$ A_k=\left[ \begin{array}{} 1 & x_1 & x_1^2 & \cdots & x_1^{n-2} & x_1^k \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-2} & x_2^k \\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^{n-2} & x_n^k \end{array}\right]. $$ If $k=n-1$, the matrix $A_{n-1}$ is called the Vandermonde matrix and the determinant of $A_{n-1}$ is given by $$ \det (A_{n-1})= \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right). $$

My question is how to prove $$ \det (A_n)= (x_1+x_2+\cdots + x_n)\prod_{1\leq i < j \leq n} \left({x_j - x_i}\right).\tag{1} $$ I know that $\det (A_n)$ is a kind of Schur polynomial, but I don't know how to apply the Schur polynomial to prove Eq. $(1)$.

Thanks in advance for your assistance.