Help differentating $f(x) = \sqrt\frac{x^2-1}{x^2+1}$ The equation I'm trying to differentiate is, $ f(x) = \sqrt\frac{x^2-1}{x^2+1}$ and I know the answer is meant to be 
$$=\frac{\frac{x\sqrt {x^2+1}}{\sqrt {x^2-1}}-\frac{x\sqrt {x^2-1}}{\sqrt {x^2+1}}}{x^2+1}$$
But when I do the working out I get this
$$=\frac{(x^2-1)^\frac{1}{2}}{(x^2+1)^\frac{1}{2}}$$ 
$$=\frac{\frac{1}{2}(x^2-1)^\frac{-1}{2}\cdot2x\cdot(x^2+1)^\frac{1}{2}-(x^2-1)^\frac{1}{2}\cdot\frac{1}{2}(x^2+1)^\frac{-1}{2}\cdot2x}{x^2+1}$$
simplify 
$$=\frac{x(x^2-1)^\frac{-1}{2}\cdot(x^2+1)^\frac{1}{2}-(x^2-1)^\frac{1}{2}\cdot x(x^2+1)^\frac{-1}{2}}{x^2+1}$$
$$=\frac{\frac{\sqrt {x^2+1}}{x\sqrt {x^2-1}}-\frac{\sqrt {x^2-1}}{x\sqrt {x^2+1}}}{x^2+1}$$
As you can see two of my $x$'s are in the wrong location, and I just can't figure out what I'm doing wrong.  Any help as to what steps I'm doing wrong or missing would be much appreciated.

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