If $p(x)$ be a polynomial satisfying $p(2 x)=p^{\prime}(x) \cdot p^{\prime \prime}(x)$, then $\sum_{x=1}^5 p(x)=$

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

$$ \begin{aligned} & \text { If } f(x)=\tan ^{-1}\left(\frac{1}{\sin ^2 x+\sin x+1}\right) \\ & \quad+\tan ^{-1}\left(\frac{1}{\sin ^2 x+3 \sin x+3}\right)+\tan ^{-1} \end{aligned} $$

$\left(\frac{1}{\sin ^2 x+5 \sin x+7}\right)+\ldots+$ upto 10 terms, then $f^{\prime}(0)=$

If $\alpha$ is such a minimum value for which the inverse of $f(x)=x^2+3 x-3$ exists in $[\alpha, \infty)$ and $g$ is the inverse of the $f$, then at $x=\alpha+\frac{5}{2}, \frac{d g}{d x}$