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

If $ y = (\tan^{-1} x)^2 $,then $ (x^2 + 1)^2 y_2 + 2x(x^2 + 1) y_1 $ is equal to

If $a y^4 = (x + b)^5$,then $\frac{y \cdot (\frac{d^2 y}{d x^2})}{(\frac{d y}{d x})^2} = $

If $f(x)$ and $g(x)$ are twice differentiable functions on $(0,3)$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$,$f^{\prime}(1)=4$,$g^{\prime}(1)=6$,$f(2)=3$,and $g(2)=9$,then $f(1)-g(1)$ is

Find the second order derivative of the function $f(x) = \log x$.

Suppose,$f(x)=e^{-\sqrt{x}}+e^{-\frac{1}{x^2}}$. If $f^{\prime \prime}(x)=\alpha \cdot \frac{e^{-\sqrt{x}}}{x}\left(1+\frac{1}{\sqrt{x}}\right)+\beta \cdot \frac{e^{-\frac{1}{x^2}}}{x^4}\left(3-\frac{2}{x^2}\right)$,then $(\alpha, \beta)=$