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

If $u = \log(\sqrt{x-1} - \sqrt{x+1})$ and $v = \sqrt{x+1} + \sqrt{x-1}$,then $\frac{du}{dv} = \dots$.

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

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

$\frac{d}{dx} \left[ \log \left\{ e^x \left( \frac{x - 2}{x + 2} \right)^{3/4} \right\} \right]$ is equal to

If $y=e^{\sin \left(\operatorname{cosec}^{-1} x\right)}$,then $\frac{d y}{d x}=$