The differential of $f(x) = \log_{e}(1 + e^{10x}) - \tan^{-1}(e^{5x})$ at $x = 0$ and for $dx = 0.2$ is

Suppose for a differentiable function $h$,$h(0)=0$,$h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$. If $g(x)=h(e^{x}) e^{h(x)}$,then $g^{\prime}(0)$ is equal to:

Find the derivative of the function given by $f(x)=\sin(x^{2}).$

$A$ possible positive value of '$a$',for which $f^{\prime}(x)=0$ has equal roots,is

$\frac{d}{dt}(\tan t + t^2 \operatorname{cosech} t)$ 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