What is a Neapolitan chord and what is its function?

A Neapolitan chord is a major chord built on the lowered second scale degree. Usually found in the first inversion, it serves a predominant function, as in a $$bII^6 -> V -> I$$ progression.

How do you analyze atonal music (music without a key?)

We often use Alan Forte's method of pitch-class sets, which describe melodies and chords using sets of numbers from 0 to 11. If the work is twelve-tone, we figure out the tone row and discover how that row was utilized to form chords and melodies. In both cases, we also look at various permutations that pop up, including augmentation, inversion, combinatoriality, and retrograde.

Explain why the derivative of the natural logarithm is $$1/x$$ using the rules of differentiation.

The derivative of $$log_b (x)$$ is the derivative of $$\frac{ln(x)}{ln(b)}$$, which equals $$\frac{1}{ln(b)} * d/dx (ln(x)) = \frac{1}{x * ln (b)}$$. When the base of the logarithm is $$e$$, then the derivative becomes $$1/x$$.