Summary: We analyze the \(s\)-dependence of solutions \(u_s\) to the family of fractional Poisson problems \[ (-\Delta)^s u = f \quad \text{in } \Omega, \qquad u \equiv 0 \quad \text{on } \mathbb{R}^N \setminus \Omega \] in an open bounded set \(\Omega \subset \mathbb{R}^N, s \in (0, 1)\). In the case where \(\Omega\) is of class \(C^2\) and \(f \in C^\alpha (\overline{\Omega})\) for some \(\alpha > 0\), we show that the map \((0, 1) \to L^\infty (\Omega), s \mapsto u_s\) is of class \(C^1\), and we characterize the derivative \(\partial_s u_s\) in terms of the logarithmic Laplacian of \(f\). As a corollary, we derive pointwise monotonicity properties of the solution map \(s \mapsto u_s\) under suitable assumptions on \(f\) and \(\Omega\). Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case \(s = 1\), i.e., for the local Dirichlet problem \(- \Delta u = f\) in \(\Omega, u \equiv 0\) on \(\partial \Omega\).