Multiplicative derivations on rank-\(s\) matrices for relatively small \(s\)

Let \(n\) and \(s\) be fixed integers such that \(n\geq 2\) and \(1\leq s\leq \frac{n}{2}\). Let \(M_n(\mathbb{K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb{K}\). If a map \(\delta:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})\) satisfies that \(\delta(xy)=\delta(x)y+x\delta(y)\) for any two rank-\(s\) matrices \(x,y\in M_n(\mathbb{K})\), then there exists a derivation \(D\) of \(M_n(\mathbb{K})\) such that \(\delta(x)=D(x)\) holds for each rank-\(k\) matrix \(x\in M_n(\mathbb{K})\) with \(0\leq k\leq s\).