What I am stupendously dumbfounded on is how the two derivatives relate to each other both qualitatively and quantitatively (in example problems).

On the one hand, (1) says that power is directly proportional to resistance, and on the other hand (2) says that power is inversely proportional to resistance. Can someone please explain to me the difference between the two? Hopefully my point of confusion is clear. Ideally a theoretical explanation and two examples. Example 1 showing why you use equation (1) and NOT equation (2), and example 2 showing why you use equation (2) and NOT equation (1).

Let me know if anything needs clarification. - gg