I've been puzzling over the best way to analyse the chords that occur at the end of the First Movement of Dvorak's Symphony No. 4 in D minor Op. 13. The passage is definitely part of a harmonic sequence, but I don't know how the C sharp major chord relates to the key of D Minor. What's the best way to analyse this chord?
There is no directly relation of C# to d minor.
As usual, in music, there doesn't need to be an explanation or rule to fit everything in. It can be quite subjective, and analyzing with tonal eyes can be confusing in fact.
To my eyes (ears?) that C# is just the dominant of the F# in the previous measure. It repeats the same pattern and motif from measures 1 and 2 in your example, jumping from tonic to dominant. First in d minor and then in F#.
I'd say this can be represented as a short modulation to F#, and then the C# is a
The appearance of the F# could be treated as a motion of whole steps from the bass or just for the effect itself that moving to a
III causes. Definitely not a IV, also your last A major is represented as IV but it should be a plain V.
The F♯ chord here is really a reference to what happened earlier in the movement. The development, after a C♯ pedal, begins in the key of F♯ major. Furthermore, several times in the exposition a C♯7 chord appears, and it's typically in the context of A major. This C♯7, which "should" resolve to F♯, doesn't; it resolves to A instead. This C♯7 even comes back enharmonically a little later as an augmented-sixth chord that moves to F.
Look through the score to get a sense of the role F♯ plays in this piece and just how often it appears.
As such, this is mostly a call-back that Dvorak is making to these prior tonal happenings. Great composers love to make these little references; it's almost as if they're like inside jokes to them.
As for understanding this F♯ within the context of this phrase, sidyll's answer is great; I'll just add a few extra details:
The F♯ in m. 3 could be understood as an extension of the D minor that began the phrase. This is because F♯ would just be scale-degree 3 in D major, so m. 3 is ultimately a stand-in for I6; here, however, it's ♯III.
The C♯ chord that comes after it serves two functions: one, it's a back-relating dominant to the F♯ that came before it. This means that we understand it as the dominant to what came before it (as opposed to a chord being the dominant of what comes after, which is much more common). The second function is due to it's resolution to A. This C♯–A pairing thus functions as a dominant to D, since the dominant of D includes both C♯ and A. (Schenkerian theorists would call this an "unfolding" of the dominant.)
Lastly, double-check your Roman numeral in the next-to-last measure. That should be a V chord, not a IV! (You labeled this chord correctly two measures earlier, so I know this is just a silly mistake.)
I'd look at bar 4 as D♭ Major. It seems to be equivalent to an augmented sixth chord in function spelled enharmonically in F major (D♭,F,A♭,B) with some weird bits. I'd say, though, that even if it's not strictly constructed like an augmented 6th chord, it still sets up the V, C major. Then the A7 substituted in is the V in D minor, the real key of the piece. It makes no sense to me to analyse the C♯ chord as ♮VII. Also, the G♭ major chord preceding it seems to be the IV of the D♭ chord, used to extend the phrase and color it.