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I have a problem with second derivatives and chain rule.

I am working on the question attached (sorry, my Latex editor wasn't working...)

I need to find F'(1) and F''(1). I managed to solve F'(1), but I can't figure out F''(1). In the second image attached, you can see the solution I saw in the source where the question came from, but I don't understand it.

I was using a tree diagram to solve F'(1), wih f going to x and y, and y going to x.

Can you help me understand this solution ?

Thank you very much.