Summary: Let \(k[[x,y]]\) be the formal power series ring in two variables over a field \(k\) of characteristic zero and let \(d\) be a non-zero derivation of \(k[[x,y]]\). We prove that if \(\text{Ker}(d)\neq k\) then \(\text{Ker}(d) = \text{Ker}(\delta)\), where \(\delta\) is a Jacobian derivation of \(k[[x,y]]\). Moreover, \(\text{Ker}(d)\) is of the form \(k[[h]]\) for some \(h\in k[[x,y]]\).