If $y = \log_3(\log_3 x)$,then $\frac{dy}{dx}$ at $x = 3$ is $\ldots \ldots$

If $y=\log_{\sin x} \tan x$,then $\left(\frac{dy}{dx}\right)_{x=\frac{\pi}{4}}$ has the value

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

The derivative of the function $f(x) = \log_5(\log_7 x)$ for $x > 7$ is:

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

$\frac{d}{d x}(\log _{|x|} e) =$ . . . . . .