If $e^{x}=y+\sqrt{y^2-1}$, then $\frac{d y}{d x}=$

If $\frac{d}{d x}\left\{\left(\frac{x-1}{x-\sqrt{x}}\right) e^{2 x+1}\right\}=\frac{x-1}{x-\sqrt{x}} e^{2 x+1} f(x)$, then $f(4)=$

$A$ 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

If $y = x \left[ \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right) + \sin x \right] + \frac{1}{2\sqrt{x}}$,then $\frac{dy}{dx} = $

Let $f$ be any function defined on $R$ and let it satisfy the condition $|f(x) - f(y)| \leq |(x - y)^2|$,for all $(x, y) \in R$. If $f(0) = 1$,then:

If $T = 2 \pi \sqrt{\frac{L}{g}}$,where $g$ is a constant and the relative error in $T$ is $k$ times the percentage error in $L$,then $\frac{1}{k} =$