Let and be differentiable functions for all

Now if , can I conclude that because is differentiable,

thus is differenitable.

And further that if is differentiable then:

and thus is twice differentiable?

If not, then how about if is bijective and thus has an inverse , and the inverse is differentiable.

Then:

And now is a combination of 2 differentiable functions, thus

then is twice differentiable?

This is not homework, I am just thinking.