myfunc[x_] := Piecewise[{{c x^2 + 2 x, x <= 2}, {x^3 - c x, True}}];
lim1 = Limit[myfunc[x], x -> 2, Direction -> -1]
lim2 = Limit[myfunc[x], x -> 2, Direction -> 1]
sol = c /. First@Solve[{lim1 == lim2}, c]

 (*2/3*)

Plot[myfunc[x] /. c -> sol, {x, 1, 3}]

D[(myfunc[x] /. c -> sol), x]