I don't like using x as the dummy variable for f^-1, particularly when the domain and range of f are not the same. f has the domain x>2, and the range f(x)>0. Write this as:
y = f(x) = sqrt(1/(x - 2)) .... where x>2 is the domain and y>0 is the range
y² = 1/(x - 2)
x - 2 = 1/y²
x = 2 + 1/y² = (2y² + 1)/y²
So f^-1(y) = (2y² + 1)/y² .... restricted to the domain y>0 so that the domain of f^-1 is identical to the range of f.
If you want to change y to x in that definition of f^-1, go ahead, but just remember it's not the same x as in f(x).
Edit: Oh yeah, the derivative. I should have left it in the first form for that, where d/dy f^-1(y) = -2/y^3 is obvious.
y = f(x) = sqrt(1/(x - 2)) .... where x>2 is the domain and y>0 is the range
y² = 1/(x - 2)
x - 2 = 1/y²
x = 2 + 1/y² = (2y² + 1)/y²
So f^-1(y) = (2y² + 1)/y² .... restricted to the domain y>0 so that the domain of f^-1 is identical to the range of f.
If you want to change y to x in that definition of f^-1, go ahead, but just remember it's not the same x as in f(x).
Edit: Oh yeah, the derivative. I should have left it in the first form for that, where d/dy f^-1(y) = -2/y^3 is obvious.