In this case I would use the 'quotient rule': \left( \frac{g}{h} \right)' = \frac{g'h - gh'}{h^2}.
This means that you have to find the derivative of $g(x) = x^2 +1$ and $h(x) = x -5$
To find $g(x)$ you can use the power rule (a specific case of the chain rule), the sum rule, and the constant rule.
First split $g(x)=x^2 + 1$ into $f(x) =x^2$ and $f(x)=1$.
The power rule is if $f(x) = x^a$ where a is a constant, then f'(x)=ax^{a-1}. So looking at $x^2$ you can see that $a = 2$ so f'(x) = 2x^{2-1} = 2x, right?
Then you take the constant rule and look at $f(x) = 1$. The constant rule is if $f(x) = a$ where $a$ is some constant, then f'(x) = 0. In this case $a=1$ and is constant so f'(1) = 0.
Now use the sum rule to put these together. The sum rule is: (f + g)' = f' + g' basically saying that you can take the derivatives separately first and then add them together. So now we take the derivatives that we already calculated and add them: 2x + 0 = 2x = g'(x).
If you now take another look at that original quotient rule, you have solved for g'... Since you already know $g$ and $h$ (without the $'$) all you need to do is find is h' and $h^2$ and plug them in...
For a good reference, it might be worth it to check out http://en.wikipedia.org/wiki/Derivative.