Please refer to attached image.

I guess just by looking at it, you can tell it's always defined and thus always differentiable.

However, I'm not precisely sure how to show this? I was thinking perhaps applying the Louiville's theorem.

Do I have to evaluate the sum, and show that it converges? I really could use some direction, as I am unsure which theorems to apply,

or further yet, how to apply them to a sum.

As for the second question, I tried applying the quotient rule, but things got really messy and I couldn't get the answer. Was this the incorrect method, or just me making a silly mistake somewhere in my working? Surely there is a way to simplify the expression.

Any help is very much appreciated!

Thanks