Derivations from the even parts into the odd parts for Lie superalgebras W and S

Let $\mathcal{W}$ and $\mathcal{S}$ denote the even parts of the general Witt superalgebra $W$ and the special superalgebra $S$ over a field of characteristic $ p>3,$ respectively. In this note, using the method of reduction on $\mathbb{Z}$-gradations, we determine the derivation space $\mathrm{Der}(\mathcal{W}, W_{\bar{1}})$ from $\mathcal{W}$ into $W_{\bar{1}} $ and the derivation space $\mathrm{Der}(\mathcal{S}, W_{\bar{1}})$ from $\mathcal{S}$ into $W_{\bar{1}}. $ In particular, the derivation space $\mathrm{Der}(\mathcal{S}, S_{\bar{1}})$ is determined.