If $y=\frac{\log _e x}{x}$ and $z=\log _e x$, then $\frac{d^2 y}{d z^2}+\frac{d y}{d z}$ is equal to

If $f(x) = e^{(x+1)^n}; (n \in N)$,then the value of $n$ for which $f''(1) = 67(2^n e^{2^n})$ is

Let $f : S \rightarrow S$ where $S =(0, \infty)$ be a twice differentiable function such that $f(x+1) = xf(x)$. If $g : S \rightarrow R$ is defined as $g(x) = \log_{e} f(x)$,then the value of $|g''(5) - g''(1)|$ is equal to:

If $y=\frac{2}{\sqrt{a^2-b^2}} \tan ^{-1}\left[\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right]$,then $\left.\frac{d^2 y}{d x^2}\right|_{x=\frac{\pi}{2}}=$

Let $f$ and $g$ be twice differentiable functions such that $f(x) \cdot g(x) = 1$ for all $x \in R$ and $f'$ and $g'$ are never zero. Then $\frac{f''(x)}{f(x)} + \frac{g''(x)}{g(x)}$ equals:

If $y = \cos^{-1} \left\{ \frac{a \cos x - b \sin x}{\sqrt{a^2 + b^2}} \right\}$,then $\frac{d^2 y}{d x^2}$ is equal to