$\frac{d}{dx} [\log(\cos x)]$

If $f(x) = \cot^{-1} \left(\frac{x^{x} - x^{-x}}{2}\right)$,then $f'(1)$ equals

If $y = \log^n x$, where $\log^n$ denotes the $n$-th iterated logarithm $\log_e(\log_e(\dots \log_e x \dots))$ ($n$ times), then $x \log x \log^2 x \log^3 x \dots \log^{n-1} x \log^n x \frac{dy}{dx}$ is equal to

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

If $y = \log_{\sin x} (\tan x)$,then $\left( \frac{dy}{dx} \right)_{\pi/4}$ is equal to

If $y=2^{ax}$ and $\left(\frac{dy}{dx}\right)_{x=1}=\log 256$,then $a=$