Make a chain of a function and its inverse: f^-1(f(x)) = x starts with x and ends with x.
Take the slope using the Chain Rule. On the right side the slope of x is 1.

Chain Rule: dx/dy dy/dx = 1 Here this says that df^-1/dy times df/dx equals 1.

So the derivative of f^-1(y) is 1/ (df/dx) BUT you have to write df/dx in terms of y.
The derivative of ln y is 1/ (derivative of f = e^x) = 1/e^x. This is 1/y, a neat slope !
Changing letters is OK : The derivative of ln x is 1/x. Watch this video for GRAPHS