If $f(x)=3 e^{x^2}$ then $f^{\prime}(x)-2 x f(x)+\frac{1}{3} f(0)-f^{\prime}(0)=$

If $f^{\prime}(x)=\tan^{-1}(\sec x+\tan x)$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$ and $f(0)=0$,then $f(1)$ is equal to:

If $f(t) = \frac{1 + \operatorname{cosec} t}{1 - \operatorname{cosec} t}$ for $0 < t < \frac{\pi}{2}$ and $f^{\prime}(t) = f(t) g(t)$,then $g(t) =$

If $f(x) = |\cos x - \sin x|$,then $f^{\prime}\left(\frac{\pi}{6}\right)$ is equal to

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

Let $f(x) = x^m$, where $m$ is a non-negative integer. The value of $m$ such that the equality $f^{\prime}(a+b) = f^{\prime}(a) + f^{\prime}(b)$ is valid for all $a, b > 0$ is