Partial derivatives of polynomial in two variables

Let $k \in \mathbb N$, $a_{ij} \in \mathbb R$ for $i,j \in \mathbb N$, $i+j \le k$. A function $f:\mathbb R^2 \to \mathbb R$

$$f(x,y) = \sum_{i+j \le k} a_{ij}x^iy^j$$

is called polynomial of degree $k$ in $x,y$.

I have to show that the partial derivatives of polynomials of degree $k \ge 1$ exist, and the partial derivatives $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$ are polynomials of degree $k-1$.

I start showing that for a polynomial of degree $1$ its partial derivatives $D_1f(x)$ and $D_1f(y)$exists and in the next step that the $D_if(x)$ and $D_if(y)$ derivative exists, but I dont know how to deal with the matrix and the indices of the sum in the formula, how can I take a take a partial derivative of it? Dealing with the matrix and other variable as a constant and use normal derivative rules for the variable in which I want to take the partial derivative?

Any help is appreciated!!!