Let the function satisfy the equation $f(x+y)=f(x)f(y)$ for all $x, y \in \mathbb{R}$,where $f(0) \neq 0$. If $f(5)=3$ and $f^{\prime}(0)=2$,then $f^{\prime}(5)$ is

$\frac{d}{d x}\left[\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^2\right]=$ . . . . . .

If the function $f(x)$ is defined by $f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^2}{2} + x + 1$,then $f'(0) = $

If $f^{\prime}(x)=a \cos x+b \sin x$ and $f^{\prime}(0)=4, f(0)=3, f\left(\frac{\pi}{2}\right)=5$,then $f(x)=$

The differential coefficient of $\frac{{\tan^{-1}x}}{{1 + \tan^{-1}x}}$ with respect to $\tan^{-1}x$ is

The differential coefficient of ${\left( {{x^{\frac{{\ell + m}}{{m - n}}}}} \right)^{\frac{1}{{n - \ell }}}} \cdot {\left( {{x^{\frac{{m + n}}{{n - \ell }}}}} \right)^{\frac{1}{{\ell - m}}}} \cdot {\left( {{x^{\frac{{n + \ell }}{{\ell - m}}}}} \right)^{\frac{1}{{m - n}}}}$ with respect to $x$ is: