Suppose

$P(x,y)=\displaystyle\sum_{i+j=0}^N \alpha_{ij}\,x^{i}y^{j}$

is a multivariable polynomial of Nth degree. How may I express the first and second derivatives of $P(x,y)$ through Mathematica?

Ps:I have read other posts, however, those did not help me.

Suppose

$P(x,y)=\displaystyle\sum^{N}_{i+j=0}\alpha_{ij}x^{i}y^{i} \equiv \alpha_{00}+\alpha_{10}x+\alpha_{01}y+a_{11}xy+\ldots+a_{0N}y^{N}$

is a multivariable polynomial of $N$th degree. I want to express the first and second derivatives of $P(x,y)$ through Mathematica.

My attemps:

Previously, I attempted to study the derivatives of the above polynomial through the following code

poly[vars_List, a_, order_] := Module[{n = Length@vars, idx, z}, idx = Cases[Tuples[Range[0, order], n], x_ /; Plus @@ x <= order]; z = Times @@@ (vars^# & /@ idx); z.((Subscript[a, Row[#]]) & /@ idx)] poly[{x, y}, a, N] (*a is used for coefficient*).

However, it looks like that this code only works when N is equal to some integer number, for example N=2.

Based on the above, is there another way to express the general form of $P(x,y)$ and obtain its first and second derivatives?

Ps:I have read other posts, however, those did not help me.

Suppose

$P(x,y)=\displaystyle\sum_{i+j=0}^N \alpha_{i}x^{i}y^{j}$

is a multivariable polynomial of Nth degree. How may I express the first and second derivatives of $P(x,y)$ through Mathematica?

Ps:I have read other posts, however, those did not help me.

Suppose

$P(x,y)=\displaystyle\sum_{i+j=0}^N \alpha_{ij}\,x^{i}y^{j}$

is a multivariable polynomial of Nth degree. How may I express the first and second derivatives of $P(x,y)$ through Mathematica?

Ps:I have read other posts, however, those did not help me.