Make Sinc'[0] return 0 instead of Indeterminate

One solution is making your own function:

MySinc[x_] := Sinc[x]
Derivative[1][MySinc] ^= 
  If[# == 0, 0, Derivative[1][Sinc] // Evaluate] &;
MySinc[0]
(* 1 *)
MySinc'[0]
(* 0 *)

And then in expressions which use Sinc use expr/.Sinc->MySinc. To me this seems like the cleanest solution. However, this can be done with Sinc, too. But it is difficult to undo!

Unprotect[Sinc];
tmp = Derivative[1][Sinc] // Evaluate;
Derivative[1][Sinc] ^= If[# == 0, 0, tmp] &;
Protect[Sinc];
Sinc[0]
(* 1 *)
Sinc'[0]
(* 0 *)

The function is:

f[x_] = D[Sinc[x], x];
f[x]
Cos[x]/x - Sin[x]/x^2

In its current form, the value at x=0 is indeterminate. It is only when taking the limit as x->0 that a value emerges. Hence:

Limit[D[Sinc[x], x], x -> 0]
0

Or, more succinctly:

Limit[f[x], x -> 0]
0