As Dr Peterson pointed out, the derivative of ln(f(x)) is not ln(f'(x))
Looking past that you tried to find the derivative of f(x) (ie what you are taking the derivative of using the quotient rule.) You did the calculus perfectly! The algebra not so good. Algebra is what kills students with Calculus!
You had -(1+ tanx)(-sec^2x) = -(1+tanx)(sec^2x). Why did -sec^2x magically become sec^2x?
(1-tanx)sec^2x - (1+tanx)sec^2x = sec^2x(1-tanx - 1 - tanx) = sec^2x(-2tanx) = -2sec^2xtanx. This is not what you got!
Subhotosh gave you the best way to do this problem.
If you have the log of ANY function, you never ever have to use the product rule, quotation rule or general power rule to compute the derivative.