Help differentating $f(x) = \sqrt\frac{x^2-1}{x^2+1}$

Going from your second last line to the last line, you put the $x$ factors into the denominator. Instead, you need to leave them in the numerator. This is because the power of $\frac{-1}{2}$ only apply to the expressions in the brackets, but the $x$ factors are outside of the brackets, so they are not affected. Thus, for example, with the first term in the numerator, you have

$$\begin{equation}\begin{aligned} x(x^2-1)^\frac{-1}{2}\cdot(x^2+1)^\frac{1}{2} & = x\left((x^2-1)^\frac{-1}{2}\cdot(x^2+1)^\frac{1}{2}\right) \\ & = x\left(\frac{\sqrt {x^2+1}}{\sqrt {x^2-1}}\right) \\ & = \frac{x\sqrt {x^2+1}}{\sqrt {x^2-1}} \end{aligned}\end{equation}\tag{1}\label{eq1}$$