I need to simplify it further, but before I do I checked it into Mathematica and it says the derivative is incorrect. What did I do wrong?
I find it to be a bit easier, though there is more factoring involved, if you divide by x first:
$\displaystyle \frac{d}{dx} \left ( - \frac{\sqrt{1 + x^2}}{x} \right ) = -\frac{d}{dx} \left ( \sqrt{ \frac{1}{x^2} + 1} \right )$
$\displaystyle = - \frac{d}{dx} \left ( \left [ \frac{1}{x^2} + 1 \right ] ^{1/2} \right ) $
Now for the derivative:
$\displaystyle = -\frac{1}{2} \cdot \left ( \frac{1}{x^2} + 1 \right ) ^{-1/2} \cdot \frac{-2}{x^3}$
$\displaystyle = \frac{1}{x^3 \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} } $
I find this result to be reasonably adequate, but your text decided differently and cleared the fraction in the denominator. So, using a little trick:
$\displaystyle = \frac{1}{x^2 \cdot x \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} } $
$\displaystyle = \frac{1}{x^2 \left ( x^2 \cdot \frac{1}{x^2} + x^2 \cdot 1 \right ) ^{1/2} } $
$\displaystyle = \frac{1}{x^2 \left ( 1 + x^2 \right ) ^{1/2} } $
-Dan