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

$g\left(x+1\right)={\log }_e\left(f\left(x+1\right)\right)={\log }_e\left(xf\left(x\right)\right)={\log }_{e}x+{\log }_e\left(f\left(x\right)\right)={\log }_{e}x+g\left(x\right)$

$$\begin{gathered} \Rightarrow g(x+1)-g\left(x\right)={\log }_{e}x \\ \Rightarrow g^{\text{'}}(x+1)-g^{\text{'}}\left(x\right)=\frac{1}{x} \end{gathered}$$

$\Rightarrow g^{\text{'}\text{'}}(x+1)-g^{\text{'}\text{'}}\left(x\right)=-\frac{1}{x^2}\dots \dots \left(1\right)$

$\text{ Putting }x=1,2,3,4\text{ in (i), we get }$

$\Rightarrow g^{\text{'}\text{'}}\left(2\right)-g^{\text{'}\text{'}}\left(1\right)=-1, g^{\text{'}\text{'}}\left(3\right)-g^{\text{'}\text{'}}\left(2\right)=-\frac{1}{4}$

$g^{\text{'}\text{'}}\left(4\right)-g^{\text{'}\text{'}}\left(3\right)=-\frac{1}{9}, g^{\text{'}\text{'}}\left(5\right)-g^{\text{'}\text{'}}\left(4\right)=-\frac{1}{16}$

$\text{ Adding. } g^{\text{'}\text{'}}\left(5\right)-g^{\text{'}\text{'}}\left(1\right)=-1-\frac{1}{4}-\frac{1}{9}-\frac{1}{16}=-\frac{205}{144}$

$\therefore \left|g^{\text{'}\text{'}}\left(5\right)-g^{\text{'}\text{'}}\left(1\right)\right|=\frac{205}{144}$