To you, t0 is a parameter. To Mathematica, it an expression which it is not sure what to do with. Subscript^(1,0)[t,0] refers to the partial derivative of this expression. You would know the partial derivative of the parameter with respect to t to be zero 0 because you know that t0 is a parameter and not some arbitrary Mathematical expression. Mathematica doesn't know that because it hasn't been told that.

Hold off on using subscripts for a while. 

If you want to learn how to properly use and assign meaning to subscripts, you'll find examples in the link in the previous post. Undesrtanding them isn't trivial and requires some work which most people begginning with Mathematica aren't ready for. 

Please. Hold off on using subscripts for a while. I've known many people who refused to wait to use them before they understood them.  Their work suffered terribly as a result.

Subscripts can have different meanings in different contexts and fields. Mathematica tries to be flexible in its interpretation of what they mean.

From the snippet you gave me, I can't be sure what the problem is. But if you are new to Mathematica, please strongly consider avoiding using subscripts for a while. Please try removing subscripts from your code and see if you are still running into problems.

Subscripted variables in Mathematica are themselves full expressions, which make them significantly different than a normal variable. You can however use the notations package to give the subscripts a meaning that will allow them to work as you want.

If you'd like, take a quick look at this list of common pitfalls for people new to Mathematica.