$\frac{d}{d x}\left(e^{\log _e \sqrt{1+\tan ^2 x}}\right) =$

If $y = \cos(x^{\circ})$ and $z = \cos x$,then $\frac{dy}{dz}$ is equal to

$A$ 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

Find the derivative of $x^{-4}(3-4x^{-5})$.

For the differentiable function $f: R - \{0\} \rightarrow R$,let $3 f(x) + 2 f\left(\frac{1}{x}\right) = \frac{1}{x} - 10$. Then $\left|f(3) + f^{\prime}\left(\frac{1}{4}\right)\right|$ is equal to

For what value of $a$ is the function $f(x) = a^x$ a strictly increasing function?