If $f(x)$ is a function such that $f^{\prime \prime}(x)+f(x)=0$ and $g(x)=[f(x)]^{2}+[f^{\prime}(x)]^{2}$ and $g(3)=8$,then $g(8)$ is equal to

If $f(x) = \frac{x^2}{x+a}$,then $f^{\prime \prime}(a)$ is equal to

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

If $y=(\tan^{-1} x)^{2}$,show that $(x^{2}+1)^{2} y_{2}+2 x(x^{2}+1) y_{1}=2$.

If $f(x)=1+nx+\frac{n(n-1)}{2}x^2+\frac{n(n-1)(n-2)}{6}x^3+\ldots+x^n$,then $f''(1)$ is equal to

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: