$\lim_{x\to0}\left(2+e^x\right)^{e^{-x}}$

$\frac{dy}{dx}=\left(\frac{y\left(x^2+3\right)}{1+2y^2}\right)$

$3y^2+3y+2y^2$

$x^2-3+\frac{3}{2}$

$\frac{x^2-2x-8}{\left(x-2\right)\left(x+2\right)}$

$\frac{d}{dx}x^3-5x^2y+2y^3=7$

$y'\left(x\right)=\sqrt[4]{-y}\:\left(x+y^2\right);\:y\left(0\right)=1$