Let f : (-1 , infinite) rightarrow R be defined by f(0) = 1 and

$f^{\prime}(x)=\frac{\frac{x}{1+x}-\ell n(1+x)}{x^{2}}$

$=\frac{x-(1+x) \ell n(1+x)}{x^{2}(1+x)}$

Suppose $h(x)=x-(1+x) \ell n(1+x)$

$\Rightarrow h^{\prime}(x)=1-\ell n(1+x)-1=-\ell n(1+x)$

$h^{\prime}(x)>0, \forall x \in(-1,0)$

$h^{\prime}(x)<0, \forall x \in(0, \infty)$

$h(0)=0 \Rightarrow h^{\prime}(x)<0 \forall x \in(-1, \infty)$

$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})<0 \forall \mathrm{x} \in(-1, \infty)$

$\Rightarrow \mathrm{f}(\mathrm{x})$ is a decreasing function for all $\mathrm{x} \in(-1$,$\infty)$