If $y=y\left(x\right)$ is an implicit function of $x$ such that ${\log }_e(x+y)=4xy$, then $\frac{d^{2}y}{ d{x}^2}$ at $x=0$ is equal to

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