Learn Turkish: Güneş sayesinde çabuk yükseldik. - Thanks to the sun, we rose quickly.

It would change the meaning from 'quickly' to 'gradually.' Saying Güneş sayesinde giderek yükseldik means 'Thanks to the sun, we rose gradually' or 'we rose more and more.'

Why is it güneş sayesinde instead of güneşin sayesinde? Doesn't sayesinde require the genitive case?

Sayesinde (thanks to) often takes the genitive case when referring to specific people or pronouns, like Ayşe'nin sayesinde (thanks to Ayşe) or onun sayesinde (thanks to him/her). However, for general concepts or natural elements like güneş, it is very common to use the bare noun. Both are grammatically correct, but güneş sayesinde feels a bit more like a fixed compound concept.

Is the verb yükseldik related to the adjective yüksek (high)?

Yes, exactly! The verb yükselmek (to rise) comes from yüksek (high). When the verb-forming suffix -el is added to yüksek, the final k drops to make pronunciation smoother. Then we just add the past tense -di and the 'we' ending -k to get yükseldik (we rose).

Could I use hızlı instead of çabuk here?