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Implicit derivation microeconomics

Hello, Im currently taking a course in economics at master level and I have a question regarding how you get to the final point in the slides I attach to this post.

I have basic knowledge of differentiation but even when googling around I cant seem to grasp how to get to the final result in the bottom. Summing up what I know about the problem:

In the indifference curve the C(L) (consumption as a function of leisure)
is introduced.

C'(L) is equal to (-1)* (The utility function differentiated wrt L divided by The utility function differentiated wrt L) or in other words = - (UL/UC)

I know that the quotient rule for differentiation is supposed to be used, so if Z=X/Y then Z'= (X'Y - XY':wink:/(Y^2) so for my problem it should probably be C''(L)=(UL'UC - ULUC':wink:/((UC)^2) and that some implicit derivation should be used to get to the final result.

Does anyone know the rules used? I cant seem to apply the rules to this problem of using the rules on already differentiated variables.

Thanks in advance!
slideslabor.png134.2KB

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