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

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

$\frac{d}{dx}(5^{\log x}) = \dots$

$\frac{d}{dx} \left[ \log \sqrt{\frac{1 - \cos x}{1 + \cos x}} \right] = $

If the curves $y = a^x$ and $y = b^x$ intersect at an angle $\alpha$,then $\tan \alpha = $

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