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Differentiate the negative-root equation $F({\beta }_{-},\delta )=0$, keeping $r$ and $\sigma$ fixed. Its derivatives are $F_{\delta }=-{\beta }_{-}$ andThus $d{\beta }_{-}/d\delta ={\beta }_{-}/F_{\beta }=-{\beta }_{-}/\Delta >0$. Differentiating the trigger ratio then givesThis is the dividend-yield sensitivity of the perpetual put trigger. A larger dividend yield lowers the risk-neutral drift of the ex-dividend stock; waiting for further price reductions becomes more attractive to the put holder, so exercise occurs at a lower price. As checks, at $\delta =0$ the negative root is $-2r/{\sigma }^2$, giving $S^{\ast }=2rX/(2r+{\sigma }^2)$. As $\delta \to \infty$, ${\beta }_{-}\to 0-$ and $S^{\ast }\to 0$.