I have the following code:
Solve[-(Sin[x]/(α*Gamma[α])) - Subscript[a, 0][x]/(α*Gamma[α]) == 0, Subscript[a, 0][x]] For replacement SetDelayed instead of Rule, I rewrite
Solve[-(Sin[x]/(α*Gamma[α])) - Subscript[a, 0][x]/(α*Gamma[α]) == 0, Subscript[a, 0][x]] /. Rule -> SetDelayed I want to calculate
Sqrt[Pi]*Derivative[2][Subscript[a, 0]][x] But I do not have any output. Any suggestions?
For better readability I substituted the subscripted variable with b
fun = -(Sin[x] / (a Gamma[a])) - b / (a Gamma[a]) // Simplify 
sol = First[b /. Solve[fun == 0, b]] 
Now use D instead of Derivative
Sqrt[Pi] D[sol, {x, 2}] 
To get the same result with Derivative you would have to write
Sqrt[Pi] (-Derivative[2][Sin][x]) which is difficult to automate
You can see why studying the Trace: 
The culprit is that what you achieve with your substitution is to evaluate Subscript[a, 0][x] := -Sin[x].
Note that this is an expression of the form f[x] := foo[x], as opposite to f[x_] := foo[x]. In other words, you did not define a function, hence why Derivative does not work.
It's the same as doing
f[x] := Sin[x] Derivative[2][f][x] which will not work.
You can use your trick, provided you also convert x into x_:
Solve[ -(Sin[x]/(\[Alpha]*Gamma[\[Alpha]])) - f[x]/(\[Alpha]*Gamma[\[Alpha]]) == 0, f[x] ][[1, 1]] // {#[[1]] /. x -> x_, #[[2]]} & // Apply@SetDelayed and then you can use Derivative as you wanted (I used f[x] instead of Subscript[a, 0][x] here, for better readability.
Of course, you don't really need to do these kinds of hacks in the first place. The kind of solution provided by eldo is a better for most circumstances.
