Question

What is the derivative of `f(x) = (1+lnx)/(1-lnx)`?

Answer 1

`f'(x) = 2/(x(1-ln x)^2)`

Explanation:

Let `t = ln x`

Then `d/dx f(x) = dt/dx d/dt (1+t)/(1-t)`

`d/dt ((1+t)/(1-t)) = 1/(1-t) + (1+t)/(1-t)^2`

`= ((1-t) + (1+t))/(1-t)^2 = 2/(1-t)^2`

Now `dt/dx = d/(dx) ln x = 1/x`

So `d/dx f(x) = 1/x 2/(1-ln x)^2 = 2/(x(1-ln x)^2)`