Derivative

It has many uses, but if the derivative of a function is greater than 0 for all x from (-inf,inf) does that mean the function itself is one to one?

yep

Yes, I'd think the best way to prove it would be to use the mean value theorem to prove the contra positive.

Take our f s.t f'(x) > 0 for all x. If there exists x and y s.t f(x) = f(y) wlog x < y

Then the mean value theorem states that there exists a z in (x,y) s.t f'(z) = [f(x) - f(y)]/x-y = 0 contradiction.


similarly you can show the function must be monotonic increasing if f' > 0, obviously both these only work with a strict inequality.

>>1

of course it does. also works if dy/dx is always < 0