If $y=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$, then $\left(1+a^2 x^2\right)y''+2 a^2 x y'$ is equal to

Suppose the function $f(x) - f(2x)$ has the derivative $5$ at $x = 1$ and derivative $7$ at $x = 2$. The derivative of the function $f(x) - f(4x)$ at $x = 1$ has the value equal to:

If $y=e^{\sin ^{-1} x}$,then $\left(1-x^2\right) y_2-x y_1=$

The function $y=e^{kx}$ satisfies $(\frac{d^2y}{dx^2}+\frac{dy}{dx})(\frac{dy}{dx}-y)=y\frac{dy}{dx}$. It is valid for

If $y=A \cos n x+B \sin n x$,then $y_2+n^2 y$ is equal to

For $n \in \mathbb{N}$,find the $n$-th derivative of $\log x$,i.e.,$\frac{d^{n}}{d x^{n}}(\log x) = $