calculus - Derivative of $frac{x}{1+x^2}$ using the limit definition.

I just started learning Calculus on my own and understand where $\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$ comes from, but I'm having trouble with this one; I think my Algebra skills are letting me down.

I start with $\lim_{h \rightarrow 0} \left(\left(\frac{x+h}{1+(x+h)^2} - \frac{x}{(1+x)^2}\right) \frac{1}{h} \right)$ but then get lost expanding everything. I don't see how to end up with $h$ as a factor in the numerator so that I can get rid of the denominator.

Answer

Hint.
$$\eqalign{\frac{f(x+h)-f(x)}{h}
&=\Bigl(\frac{x+h}{1+(x+h)^2}-\frac{x}{1+x^2}\Bigr)\frac{1}{h}\cr
&=\frac{(x+h)(1+x^2)-x(1+(x+h)^2)}{h(1+x^2)(1+(x+h)^2)}\cr
&=\frac{1+x^2-2x^2-hx}{(1+x^2)(1+(x+h)^2)}\cr}$$
and now it is easy to take $h\to0$.