Question

find the derivative of ln(sqrt((1-ln(x))/(1+ln(x)))

Answer 1

It looks messy, but you have
y = ln(sqrt((1-ln(x)) - ln(1+ln(x)))
= 1/2[ ln(1-lnx) - ln(1+lnx) ]
Now, using the chain rule, that is
y' = 1/2 [ 1/(1-lnx)*(-1/x) - 1/(1+lnx)*(1/x) ]
= 1/2 * 1/(x(lnx-1)(lnx+1))

Answer 2

In google paste:
math10 derivative calculator

Whe you see list of results click on:
Free Step-by-Step First Derivative Calculator(Solver) - Math10

When page be open in rectangle Function paste:

ln(sqrt((1-ln(x))/(1+ln(x))))
and click option:
Find derivative

You will see solution step-by-step.