I first started wanting to know about the derivation of theorems because certain ones help you memorize the theorems better.
But as I take harder math classes, it turns out better for me to use other mnemonic tricks. But I still want to know about proof and derivation because I guess I am curious and feel uncomfortable with something that comes out of nowhere.
But as some people ask for to prove things for them, I start to notice I often forgot the proofs.
It usually happen like this: I can write down the first few steps of the proof. (At least I can start it) But I often got stuck and don't know where to continue from.
Take for example, this simple proof of logarithm change of base formula, I saw on wiki proof: let $$y=\log_b x \iff b^y=x$$ $$z=\log_a x \iff a^z=x$$ Then, $$z=\log_a (b^y)$$ $$=y\log_a b$$ $$=\log_b x \log_a b$$
I read this like 1 week ago. Today , I try reciting this proof. I got the first 2 line down on paper. (The "let" conditions) Then I got stuck and had to go back on to the site again.