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In order to get the expression yo simplify to the fullest extent, one could make the additional assumption that the label $\alpha$ always falls within the range of the summation whose index is $\beta$. To do this, we have to add some rules which are easier to write if the starting expression is brought into a slightly different form: $$\frac{\partial ...


D[f[g[t]*h[t]], t] /. t -> 0 Derivative[1][f][ g[0] h[0]] (h[0] Derivative[1][g][0] + g[0] Derivative[1][h][0]) EDIT: I misread your post. Same approach D[f[g[t], h[t]], t] /. t -> 0 Derivative[1][h][0]* Derivative[0, 1][f][g[0], h[0]] + Derivative[1][g][0]* Derivative[1, 0][f][g[0], h[0]]

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