Na de bus voorrang te hebben gegeven, rijdt hij rustig verder.

Breakdown of Na de bus voorrang te hebben gegeven, rijdt hij rustig verder.

de bus
the bus
geven
to give
rustig
quiet (calm, peaceful)
de voorrang
the right of way

Questions & Answers about Na de bus voorrang te hebben gegeven, rijdt hij rustig verder.

Why is it te hebben gegeven instead of just te geven?
The structure na ... te hebben plus a past participle (like gegeven) means 'after having [done something]'. Because giving way happened before he drove on, we use this perfect infinitive form to show a completed action.
Why does te hebben gegeven come at the very end of the first clause?
In a na ... te clause, the verb parts must go at the very end. All the other details in that clause (de bus voorrang) get sandwiched between Na and te hebben gegeven.
Why is the word order rijdt hij instead of hij rijdt?
The entire first clause (Na de bus voorrang te hebben gegeven) acts as one big time phrase at the start of the sentence. Because of the V2 (verb second) rule in Dutch, the main verb (rijdt) must immediately follow this phrase, pushing the subject (hij) to the third position.
Where did the preposition aan (to) go? Shouldn't it be giving way 'to' the bus?
You can definitely say voorrang aan de bus, but Dutch often drops the preposition aan if the receiver of the action (de bus) is placed right before the direct object (voorrang).
What is the function of verder at the end of the sentence?
It belongs to the separable verb verderrijden, which means 'to drive on' or 'to continue driving'. In a normal main clause like this, the verb splits up: the conjugated part (rijdt) takes the second position, and the separable prefix (verder) goes to the very end.

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