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I think most upper level analysis books (including Rudin) decide to "reintroduce" the idea of a derivative by transitioning from measure theory to the idea of differentiating measures.

I get that this theory has power because it extends to a lot more than "differentiable functions" as we learn in high school, but specifically what are the advantages of this concept of differentiating measures (as opposed to functions)?

Also, one extra minor question. What's the quickest way to recover the standard $$lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ definition of a limit from the derivative of measures lingo?

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