I've been reading the Michael Spivak's book on calculus lately and one thing that is bothering me is how he renders out proofs for various theorems without spending some time on the "golden facts" of how it works or the historical derivation of such a rule.
$\frac{df(x)}{dx} = nx^{n-1}$
I am currently searching all over the place to find a bit more how we came from the limit definition to this simple polynomial expression. Could someone shed some more light on this or recommend a book that delves deeper into the core of these mechanics? I'm not interested in just proofs of theorems, I'd like to explore how and why they work. I have no doubt that they work, so I'd more appreciate the why/how than some proof anyone can do.
This is exactly the reason I wanted to explore this book, unfortunately, it doesn't satisfy my particular curiosity regarding derivatives/differentiation. Everywhere I try, it seems like everyone is trying to avoid that particular connection I personally deem quite important. Like everyone just learned how to use them, with little care about how or why it works.
Perhaps it's obvious or inferred, I don't know. Please help.