Clear[f, g, simplelist, d, dlist1, dlist2]
f[x_] := Exp[ -x^2]
g[n_] := f[x](-1)^n
simplelist = g[{0, 1, 2, 3, 4}]


d[n_] := Derivative[n][f][x] 
dlist1 = d[{0, 1, 2, 3, 4}]
dlist2 = {d[0],d[1], d[2], d[3], d[4]}