Hint: Use the chain rule for differentiation. If you have say z = (y*dy/dx) then use the fact that if y = uv, then y' = vu' + uv'.
Basically use the intuition of the differentiation results (chain rule, quotient, power rule, etc) and apply them as needed. If you have a function that is not well defined, then the intuition for the results becomes more important because it has more utility than trying to make sense of something that is more abstract.
Once you let the math be intuition, the seemingly harder and abstract problems will make more sense in this context.