We give explicit formulas for the subdifferential set of the conjugate of not necessarily convex functions defined on general Banach spaces. Even if such a subdifferential mapping takes its values in the bidual space, we show that, up to a weak(auau) closure operation, it is still described by using only elements of the initial space relying on the behavior of the given function at the nominal point. This is achieved by means of formulas using the epsilon-subdifferential and an appropriate enlargement of the subdifferential of this function, revealing a useful relationship between the subdifferential of the conjugate function and its part lying in the initial space.