This derivation takes up a lot of space.
Many years ago, when I did it on the test, it took up around three sheets of paper to get everything,
but I derived it properly, and I earned extra credit on the bonus problem.
Notice how I always back up and think things through while simultaneously moving forward.
Some students would try to work all the way through and then come back and check the problem for errors, and that's just fine for simpler problems. However, on long processes, it often causes the student to forget their place, become confused, and give up. I developed this "double check while you work method," and it is very effective. I like the fact that I made some errors, and then went back and corrected them before moving forward. It's a good training experience.
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