f[a_, t_, c_] = Sum[c[k2] * exp[a, t[k2]], {k2, 1, 3}]

f = Sum[c[k] Exp[-(a - t[k])^2], {k, kmax}]

(* 2 E^-(a - t[n])^2 c[n] (a - t[n]) *)

Assuming [kmax > 3, D[f, t[3]]]
(* 0 *)

D[f, t[k]]
(* Sum[(2*c[k]*(a - t[k]))/E^(a - t[k])^2, {k, kmax}] *)

D[f[[1]] /. k -> n, t[n]]
(* 2 E^-(a - t[n])^2 c[n] (a - t[n]) *)

f = Sum[c[k] Exp[-(a-t[k])^2], {k, kmax}];

Assuming[i ∈ Integers && 1<=i<=kmax, Simplify @ D[f, t[i]]]