The inverse function theorem says that if y = f(x) has an inverse x = g(y), then then g'(y) will be 1/f'(x). There is an easy way to see this (though it is much more difficult to prove!):

If we have a function y(x), then dx/dy = 1/(dy/dx). You can "clear the fraction" in the "denominator" and see that the RHS is also dx/dy. By construction of the derivative, dx/dy would be the derivative of the inverse function.

Before someone more Mathematically minded than me comes along to remind you, let me say that treating dy/dx as a fraction is averytricky process and is not a strictly legitimate process even where you can get away with it. (But it does occasionally provide a convenient guideline.)

-Dan