Why is a first/second derivative useful in spectroscopy?
Derivative spectroscopy was at its peak in the 70-80s. It is still very useful. The only problem with derivatives is that they enhance noise like crazy i.e., they suppress low frequency signals and enhance high frequency signals. 

Before you do anything with the derivatives, you have to apply a smoothing process on them, specially on the second derivative. This "problem" of noise with derivatives was very well known then and smoothing was necessary before anyone used them.

For smoothing, you have apply a filter in such a way that it does not shift the peak positions, and the simplest one is the centered moving average. Choose an odd number of data points, say 5 (or higher), so the data at point x_3 will have two data points above and two data points below. Take the average of all 5, at position number x_3. Continue doing this for the rest of the points. You will start to see features in the second derivative.

Basically, derivatives are used to reveal the fine structure of a band. Shoulders become clear. The second derivative can "act" like a high resolution signature of a given molecule.

A very nice paper on higher order derivatives is 
High‐Resolution, Higher‐Order UV/VIS Derivative Spectrophotometry

And there are some other tricks with second derivatives to resolve overlapping signals by my co-author.

https://terpconnect.umd.edu/~toh/spectrum/Differentiation.html