I have done 2 of the 3 parts of the following problem but now I'm stuck. Any help would be appreciated:

Let . Suppose is differentiable at and for all .

1)Prove

2)Prove

3)Prove is differentiable on , and .

I have already proven parts 1 and 2. For part 3 I'm guessing you set it up something like this... and then you use the fact that to rearrange to find the derivative but I'm struggling to see how. Any help would be great.