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For a simple counterexample, consider $X =[0,1]$. If the map $t \mapsto \delta_t$ were differentiable in either sense then for every bounded linear functional $F$ on $AE(X)$ the map $t\mapsto F(\delta_t)$ would be differentiable. Recalling that the dual of $AE(X)$ is ${\rm Lip}_0(X)$, differentiability at $t$ would imply that every Lipschitz function on $[0,1]$ is differentiable at $t$, which is obviously false.