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

If $y = \cos^{-1}(\tanh x) + \sinh(\sin 6x)$, then $\frac{dy}{dx} =$

Let $g(x) \neq 0, g^{\prime}(x) \neq 0, f(x) \neq 0, f^{\prime}(x) \neq 0$. If $F(x)=f(x) g(x)$, $G(x)=f^{\prime}(x) g^{\prime}(x)$, $F^{\prime}(x)=G(x) H(x)$, and $F^{\prime}(x)=F(x) K(x)$, then $H(x)+K(x)=$

Find the derivative: $\frac{d}{dx} \left( \frac{e^x}{1 + x^2} \right)$

If $y = \frac{e^{\sin x} + \sinh^3 x}{\cosh x - \tan x}$, then $y^{\prime}(0) = $

If $f(x) = \sin([\pi^{2}]x) + \cos([-\pi^{2}]x)$,then find $f'(x)$,where $[\cdot]$ denotes the greatest integer function.