The derivative of $\log |x|$ is

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

If $y = \log \left(\frac{1-x^{2}}{1+x^{2}}\right)$,then $\frac{dy}{dx}$ 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

If $y = \log_{\cos x} \sin x$,then $\frac{dy}{dx}$ is equal to

The derivative of $(\log x)^{\sin x}$ with respect to $\cos x$ at $x=\frac{\pi}{2}$ is