\frac{d}{dx}(ln(\frac{x}{x^{2}+1})) \\ \\ \text{When differentiating logs, it's just} \frac{d}{dx}ln(f(x)) = \frac{f`(x)}{f(x)} \\ \text{So first we differentiate} \frac{x}{x^{2} + 1}, \text{ using the quotient rule, which gives us:} \frac{1-x^{2}}{({1+x^{2})^{2}}{}} \\ \\ \text{And then we...