If $y = \sin(\log_e x)$, then $x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx}$ is equal to

If $y = a^x \cdot b^{2x-1}$,then $\frac{d^2 y}{d x^2}$ is equal to

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 $(\alpha, \beta)=$

If $y(\theta) = \frac{2 \cos \theta + \cos 2 \theta}{\cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2}$,then at $\theta = \frac{\pi}{2}$,$y'' + y' + y$ is equal to:

If $y=\log \left(x-\sqrt{x^2-1}\right)$, then $\left(x^2-1\right) y^{\prime \prime}+x y^{\prime}$ is equal to:

Let $\cos ^{-1}\left(\frac{y}{b}\right)=\log \left(\frac{x}{n}\right)^{n}$. Then