The derivative of $\log _{e^2}(\log x)$ with respect to $x$ is $ . . . . . . $.

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

If $f(x) = \log_{5} \log_{3} x$, then $f^{\prime}(e)$ 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 the curves $y = a^x$ and $y = b^x$ intersect at an angle $\alpha$,then $\tan \alpha = $

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