Quotient Rule of Differentiation

Q. Suppose $f$ and $g$ are functions having second derivative $f\text{'}\text{'}$ and $g\text{'}\text{'}$ respectively at all points. If $f\left(x\right)\cdot g\left(x\right)=1$ for all $x$ and $f\text{'}$ and $g\text{'}$ are never zero, then $\frac{f\text{'}\text{'}\left(x\right)}{f\text{'}\left(x\right)}-\frac{g\text{'}\text{'}\left(x\right)}{g\text{'}\left(x\right)}$equals

1. $\frac{-2f^{\prime }\left(x\right)}{f\left(x\right)}$
2. $\frac{-2g^{\prime }\left(x\right)}{g\left(x\right)}$
3. $\frac{-2f^{\prime }\left(x\right)}{f\left(x\right)}$
4. $\frac{2f^{\prime }\left(x\right)}{f\left(x\right)}$

Q. Let $f:(-\frac{1}{\sqrt{2}},1]\to (-\infty ,\ln \sqrt{2}]$ be a function defined as $f\left(x\right)=\ln (x+\sqrt{1-x^2})\text{and}g\left(x\right)=\frac{x^2}{f\left(x\right)}$ .
If $f\left(x_0\right)=\ln \sqrt{2},$ then

1. $g\left(x_0\right)=\frac{1}{\sqrt{2}\ln 2}$
2. $g\left(x_0\right)={\log }_{2}e$
3. $g^{\prime }\left(x_0\right)=2\sqrt{2}{\log }_{2}e$
4. $g^{\prime }\left(x_0\right)=\sqrt{2}{\log }_{e}2$

Q. Suppose $f$ and $g$ are differentiable functions on $(0,\infty )$ such that $f\text{'}\left(x\right)=-\frac{g\left(x\right)}{x}$ and $g\text{'}\left(x\right)=-\frac{f\left(x\right)}{x}$, for all $x>0$. Further, $f\left(1\right)=3$ and $g\left(1\right)=-1$.

If $f\left(x\right)-g\left(x\right)=B{x}^m,$ for all $x>0$ and some constant $B$, then the value of $m$ equals

1. $1$
2. $2$
3. $1$
4. $-2$

Q. If P$(x_1,y_1)$, Q$(x_2,y_2)$, R$(x_3,y_3)$ be the points of inflection of the curve $x^{2}y-x+y-1=0,$ then number of rational vertices of the polygon PQR is