I'm just not getting the pronunciation of cueillir and the derivatives. The "ill" disappears.

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Accepted Answer

Bonjour Julia,
You have to think of ueill as a combination: oo (as in "cook") followed by a y sound. So cueillir is pronounced [koo yir].
This is a variation of the letter combination euill pronounced the same way, as in feuille. The e and u change places because of the c.
You might find this helpful: 
https://www.lawlessfrench.com/pronunciation/ueil/

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Hi Julia - Cueillir isn't spelled the way it sounds. It's more like "kuh-yir". Hope that helps!

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That’s what I thought but the audio sounds more like Kay to me.

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The double L's in French are sometimes pronounced as an L sound (as in English) or as a Y sound (as in Spanish).  Here the "lle" is like the 'y' or 'ya' sound like in viellle or portefeuille.