Let $X$ and $Y$ be elliptic curves (over an algebraically closed field, but no assumptions on the characteristic) with Jacobians $J_X$ and $J_Y$ respectively. Suppose $f:X\to Y$ is an isogeny, with dual isogeny $\widehat{f}:J_Y\to J_X$. How can I show that the map on tangent spaces $d\widehat{f}:T_0J_Y\to T_0J_X$ is $f^*:H^1(Y,\mathcal{O}_Y)\to H^1(X,\mathcal{O}_X)$? I already know that the tangent space to the identity of $J_X$ is $H^1(X,\mathcal{O}_X)$ via considering maps $\operatorname{Spec} k[\varepsilon]/\varepsilon^2\to J_X$ and using the universal property of the Jacobian variety, but I'm a little stumped on how to rigorously show the statement about the map.
Background: I'm trying to connect the two characterizations of the Hasse invariant of an elliptic curve in terms of the action of the Frobenius on $H^1$ and the separability of the dual of the Frobenius. Knowing this statement would finish the problem by the link between separability and the map on tangent spaces (plus maybe an argument that the map on tangent spaces has to be the same everywhere? I'm actually realizing I might need a little help with that too, as I'm writing this).
I later asked this at MathOverflow and got a hint from Piotr Achinger to use the fact that pullback on line bundles agrees with pullback on $H^1(\mathcal{O}^\times)$ and apply this with the naturality of the exact sequence $0\to \mathcal{O}\to\mathcal{O}[\varepsilon]\to\mathcal{O}^\times\to 0$ to get that the induced map on $H^1$ is the same as the map on tangent spaces of the Jacobian.