Can we define tangent spaces for $C^k$ manifolds via derivations or cotangent spaces?

Question

Wikipedia gives three definitions of the tangent space of a Manifold:

• Via tangent curves,
• Via derivations, and
• Via cotangent spaces.

For the last two, the manifold is assumed to be $C^{\infty}$, as opposed to the definition via tangent curves for which the manifold can be $C^k$.

Can either of the last definitions be generalized to $C^k$ manifolds?