For \(n\leq 3\) it was proved by Nagata and Nowicki that the kernel of a derivation on \(K[X_ 1,\dots,X_ n]\) is of finite type over \(K\). This paper proves that for large values of \(n\) this is no more true. Namely using Nagata's counter-example to Hilbert's fourteenth problem which gives a subring \(R=L \cap\mathbb{C}[X_ 1,\dots,X_ n]\) not of finite type where \(L\) is a subfield of \(\mathbb{C}(X_ 1,\dots,X_ n)\), the author constructs a derivation of \(K[X_ 1,\dots,X_ r,Y_ 1,\dots,Y_ r]\) for \(r\geq 4\) such that the kernel of this derivation is exactly the ring \(R\) as in Nagata's counter-example hence not of finite type over \(K\). The main idea is that the kernel of the derivation given by \(D=\partial/\partial X_ 1+ X_ 2(\partial/\partial X_ 2)+\dots+ X_ 2\cdot \dots \cdot X_ n(\partial/\partial X_ n)\) is exactly \(K\). This is also refined to show that if \(L\) is a field extension of \(K\) of transcendence degree \(n\) such that \(K\) is algebraically closed in \(L\) then there exists a derivation of \(L\) with kernel \(K\).