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

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

$\frac{d}{dx}(e^{x \log x} + e^3) = $ . . . . . .

If $f(x) = \log_{(x^2-2x+1)}(x^2-3x+2)$, $x \in R - \{1, 2\}$ and $x \neq 0$, then $f'(3) =$

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

If $y = \log_{2026}(\log_{2025} x)$, then $\frac{dy}{dx} = \dots \dots \dots$