If $f(x) = \log_{5} \log_{3} x$,then $f^{\prime}(e)$ 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

Let $f(x)=e^x$,$g(x)=\sin^{-1} x$ and $h(x)=f(g(x))$,then $\frac{h'(x)}{h(x)}$ is equal to

$\frac{d}{dx} \left( \log \sqrt{\frac{1+\sin x}{1-\sin x}} \right) = $

If $f(x) = \cot^{-1} \left(\frac{x^{x} - x^{-x}}{2}\right)$,then $f'(1)$ equals

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