On the algebra of constants of polynomial derivations in two variables

Let d be a k-derivation of k[x,y], where k is a field of characteristic zero. Denote by ${\displaystyle w\tilde{d}}$ the unique extension of d to k(x,y). We prove that if ker d ≠ k, then ker ${\displaystyle w\tilde{d}}$ = (ker d)_0, where (ker d)_0 is the field of fractions of ker d.