It's very dangerous to combine steps! Until you've really got the hang of it, write out your calculations with at most one rule per line. For instance, you might write:
\begin{eqnarray*}F'(x) = & \frac{(1+e^{2x})\frac{d}{dx}\Big(\sin\left(x^2\right)\Big) - \sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by the quotient rule)} \cr= & \frac{\left(1+e^{2x}\right)\left(\cos\left(x^2\right)\right)(2x) - \sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by the chain rule applied to }\sin\left(x^2\right)\hbox{)} \cr= & \frac{\left(1+e^{2x}\right)\left(\cos\left(x^2\right)\right)(2x) - \sin\left(x^2\right)e^{2x}\frac{d}{dx}\Bigl(2x\Bigr)}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by the chain rule applied to }e^{2x}\hbox{)} \cr= & \frac{2x\left(1+e^{2x}\right)\cos(x^2) - 2\sin\left(x^2\right)e^{2x}}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by algebra)} \cr\end{eqnarray*}