Product Rule of Differentiation

Q. Let $f\left(x\right)$ and $g\left(x\right)$ are polynomial of degree 4 such that
$g\left(\alpha \right)=g^{\prime }\left(\alpha \right)=g^{\prime \prime }\left(\alpha \right)=0$.
If $\underset{x\to \alpha }{lim}\frac{f\left(x\right)}{g\left(x\right)}=0,$ then number of different real solutions of equation $f\left(x\right)g^{\prime }\left(x\right)+g\left(x\right)f^{\prime }\left(x\right)=0$ is equal to

Q. If $y^x-x^y=1$ then $\frac{dy}{dx}$ at $x=1$ is

1. $2(1-\ln 2)$
2. $2(1+\ln 2)$
3. $\ln \left(4e^2\right)$
4. $\ln \left(\frac{e}{4}\right)$