Originally Posted by
Turiski EDIT: To clarify, I am answering part 2.
When dealing with derivatives, the most important thing is to be careful. There are dropped and confused signs, arithmatic errors, and your substitution for v is wrong. (Also, not that it matters, but you kind of pulled your last step out of nowhere: continuing from the step above would give 2x). I'm sorry to say but I have to scrap everything except the mini-derivatives; those were correct.
u'v + v'u =
$\displaystyle (1-\frac{1}{x^2})(x-\frac{1}{x}+1) \, \, + \, \, (1+\frac{1}{x^2})(x+\frac{1}{x})$
$\displaystyle = (x-\frac{1}{x}+1) \, - \, (\frac{1}{x}-\frac{1}{x^3}+\frac{1}{x^2}) \, + \, (x+\frac{1}{x}) \, + \, (\frac{1}{x}+\frac{1}{x^3})$
$\displaystyle x-\frac{2}{x}+1+\frac{1}{x^3}-\frac{1}{x^2}+x + \frac{2}{x} + \frac{1}{x^3}$
$\displaystyle 2x+1-\frac{1}{x^2}+\frac{2}{x^3}$