The power rule and the chains rule

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Homework Statement



How do I know for sure when to use the power rule instead of the chain rule and vice versa?

Homework Equations





The Attempt at a Solution

 
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power rule is used when there are no variables in the raised part (ie. x^2 or 3x^6) but chain rule is used if there are variables in the raised part (ie. d/dx (e^(2x))=e^(2x) * d/dx (2x) = 2e^(2x) Chain rule is used in other situations but this is basically the situation where you would have to decide between chain and power rule as you are asking.
 
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Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...

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