One of my first times using mathematica, I'm getting a weird Subscript(1,0) output when I try to solve a derivative. What is going on and how can I fix this?

One of my first times using mathematica, I'm getting a weird Subscript(1,0) output when I try to solve a derivative. What is going on and how can I fix this?

The reason is, that the nicely formatted subscripts in the frontend looking like $c_f$ are no "real" variables, but are represented as Subscript[c,f]. You can see that, if you use FullForm on one of those subscript-lookalikes.
Therefore, your code does end up trying to create the derivative of the Mathematica-function Subscript, but of course there is no such thing. This is why you are left with that result.
Solution:
Exchange any subscript with a more straightforward representation, e.g. (as suggested by Marius): cf instead of Subscript[c,f] etc.
D[Subscript[c, t], c]. Just usecf,nfas variable names instead. – Marius Ladegård Meyer Mar 21 '15 at 17:54