Derivative of a log-quotient using logarithm properties

First rewrite $f(x)$ using the quotient property of logarithms: $f(x) = \ln(x+1) - \ln(x-1).$ This avoids needing the quotient rule and turns the problem into differentiating two simple log expressions.

Now differentiate term by term using $d/dx[\ln u] = u'/u$: $f'(x) = \frac{1}{x+1} - \frac{1}{x-1}.$ Combining over a common denominator gives $f'(x) = \frac{(x-1)-(x+1)}{(x+1)(x-1)} = \frac{-2}{x^2-1}.$

Rewriting ln of the quotient as a difference of two logarithms let you avoid the quotient rule and the chain rule complications from differentiating a fraction directly. Instead of applying the quotient rule to the fraction inside the logarithm, splitting it turned the problem into two simple logarithm derivatives. This is why the solution explicitly notes it avoids needing the quotient rule.