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 $\dfrac{dy}{dx} = \ldots$

Let $y = \sqrt[p]{x^3 y}$. If $\dfrac{dy}{dx} = \dfrac{3}{2}$ when $y = 1$, then the value of $p$ is equal to $\ldots$