If $y=e^{\log _{e}\left[1+x+x^{2}+\ldots\right]}$,then $\frac{d y}{d x}$ is equal to

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

Differentiate with respect to $x$ the following function: $\log _{7}(\log x)$

$\frac{d}{dx}(\log \tan x) = $

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

If $f(x) = \log_{x^2}(\log x)$,then at $x = e$,$f'(x)$ has the value