The derivative of $x^{5x}$ with respect to $x$ is

if $x=\sqrt{a^{\tan^{-1}t}}$ and $y=\sqrt{a^{\cot^{-1}t}}$ then which one of the following is true?

If $y = \left(x + \sqrt{x^2+1}\right)^m$, then $\frac{dy}{dx}$ is

If $y=7^x\log(5x^3)$, then $\dfrac{dy}{dx}$ is equal to: (consider $\log_e x = \log x$)

If $x = e^{y + e^{y + e^{y + \ldots \infty}}}$, $x > 0$, then $\frac{dy}{dx}$ is equal to

If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:

If $y = x\log_e\left(\dfrac{x}{2+x}\right)$, then $x^3\dfrac{d^2y}{dx^2}$ is equal to: