If $f(x)=\log _{x^{2}}\left(\log _{e} x\right)$,then $f^{\prime}(x)$ at $x=e$ is

If $y = \log \left( \frac{1 + \sqrt{x}}{1 - \sqrt{x}} \right)$,then $\frac{dy}{dx} = $

If $y = \log_{\sin x} (\tan x)$,then $\left( \frac{dy}{dx} \right)_{\pi/4}$ is equal to

If $f(x)=\log _e\left(e^{2 x}\left(\frac{3 x+5}{5-3 x}\right)^{\frac{2}{3}}\right)$,$x \neq \frac{-5}{3}, \frac{5}{3}$,then the value of $\frac{d f}{d x}$ at $x=1$ is

Consider the following statements:
Statement $1$: If $y = \log_{10} x + \log_{e} x$,then $\frac{dy}{dx} = \frac{\log_{10} e}{x} + \frac{1}{x}$.
Statement $2$: $\frac{d}{dx}(\log_{10} x) = \frac{\log x}{\log 10}$ and $\frac{d}{dx}(\log_{e} x) = \frac{\log x}{\log e}$.

Let $\ln x$ denote the logarithm of $x$ with respect to the base $e$. Let $S \subset R$ be the set of all points where the function $\ln(x^2-1)$ is well-defined. Then,the number of functions $f: S \rightarrow R$ that are differentiable,satisfy $f^{\prime}(x)=\ln(x^2-1)$ for all $x \in S$ and $f(2)=0$,is