a variation on the chain rule

Question

Consider the composition of functions $A = B \circ C$ and choose 2 of the functions to be differentiable, so there are $\binom{3}{2}$ cases

1. (Chain rule) If B and C are differentiable, then A is differentiable.
2. If A and B are differentiable, is C differentiable?
3. Likewise, if A and C are differentiable, is B differentiable?

The only way I see so far is to use the inverse function theorem, but I feel this is not the simplest approach, plus it introduces the additional hypothesis of requiring non zero derivative. So is there a better approach?