Let $x = at^2 - 1$, where $a > 0$ and $y = t^3 + 1$. If at $t = 1$, $\dfrac{d^2y}{dx^2} = \dfrac{3}{16}$, then the value of $a$ is...

If $y = \cot^{-1}\left(\dfrac{1 + \sin 5x}{\cos 5x}\right)$, then the value of $\dfrac{dy}{dx}$ is

The derivative of $\log_{10} x$ with respect to $\log_x 10$ is

For $x > 0$ and $(x \log x) < 1$, if $y = \cot^{-1}\left(\dfrac{x - \log x^{x^2}}{\log e^{x^2} + \log x^x}\right)$, then $\dfrac{dy}{dx} = \ldots$