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

If $\frac{d}{dx} \left( A \log \left( \frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1} \right) \right) = \frac{1}{x \sqrt{1-x^3}}$,then $AB=$

$\frac{d}{dx} \left\{ \log \left( \frac{e^x}{1 + e^x} \right) \right\} = $

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

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

For $x < 0$,$\frac{d}{dx} [|x|^x] = $