how can i proof g'(x)=1/F'(G(X)) if F(X) &G(X) are inverse of each other?

Question

how can i proof g'(x)=1/F'(G(X))   if F(X) &G(X) are inverse of each other?

dumbcow · Answer

http://en.wikipedia.org/wiki/Inverse_function#Inverses_and_derivatives

psymon · Answer

Well, in terms of a proof, how about this:
If $$g(x)=f ^{-1}(x)$$ Then let y = f^-1(x)
Therefore f(y) = x and f'(y) = dx/dy. Since y = g(x), we can say
f'(g(x)) = dx/dy. And if g(x) = y, then g'(x) = dy/dx, which is the same as:
$$\frac{ 1 }{ (dx/dy) }=g'(x)$$
If that makes sense o.o

dumbcow · Answer

haha im thinking how confusing that ^^ looks to a beginning calc student

psymon · Answer

I don't know the best way to like....dumb it down o.o

psymon · Answer

I would imagine the wiki page on it would be more confusing, though, lol.

dumbcow · Answer

me neither, it takes me a few minutes of just staring at it