$1)f'(x)=f^{-1}(x)\\ 2)g'(x)=g(-x)\\ 3)h(x)=h'(h(x))$
I've solved these and you can see the solutions HERE. What I'm wondering is what would I call these when describing them to people? I'm calling them 'odd equations' right now but I'd love to know if there's a legitimate name for them. The closest thing to them is differential equations so that's the tag I'm using.
Any help would be greatly appreciated, I'd love to find more of these.
I suggest differential functional equations (or possibly functional differential equations).
Functional equations typically have terms like $f(x+y)$ or $f(f(x))$, but I've rarely seen ones with derivatives. So to specify that derivatives appear (and thus implicitly assume that the solutions are differentiable, which is similarly uncommon), that's what I would call them.