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

If $y = \log_3(\log_3 x)$,then $\frac{dy}{dx}$ at $x = 3$ is $\ldots \ldots$

If $y=2^{\log x}$,then $\frac{d y}{d x}$ is

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

If the curves $y = a^x$ and $y = b^x$ intersect at an angle $\alpha$,then $\tan \alpha = $

If $y=\log \sqrt{\frac{1+\sin x}{1-\sin x}}$,then $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$ is