If $f(2)=4$ and $f^{\prime}(2)=1$,then $\lim _{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}$ is equal to

If $f(x) = \sin \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right)$, then $f^{\prime}(1) = $

The derivative of $e^{ax} \cos bx$ with respect to $x$ is $re^{ax} \cos(bx + \alpha)$,where $\alpha = \tan^{-1}(\frac{b}{a})$. When $a > 0, b > 0$,the value of $r$ is:

Find the derivative: $\frac{d}{dx} \left[ \log \left( x + \frac{1}{x} \right) \right] = $

If $f(x) = \sqrt{x^2 + 1}$, $g(x) = \frac{x + 1}{x^2 + 1}$, and $h(x) = 2x - 3$, then find the value of $f'(h'(g'(x)))$.

$A$ particle starts from rest and its angular displacement (in radians) is given by $\theta = \frac{t^{2}}{20} + \frac{t}{5}$. If the angular velocity at the end of $t = 4 \ s$ is $k$,then the value of $5k$ is