$2y\frac{{dy}}{{dx}} = \frac{{(1 - {e^x}){e^x} + (1 + {e^x}){e^x}}}{{{{(1 - {e^x})}^2}}} = \frac{{2{e^x}}}{{{{(1 - {e^x})}^2}}}$
$\therefore \frac{{dy}}{{dx}} = \frac{{{e^x}}}{{{{(1 - {e^x})}^2}}}\sqrt {\left[ {\frac{{1 - {e^x}}}{{1 + {e^x}}}} \right]\left[ {\frac{{1 - {e^x}}}{{1 - {e^x}}}} \right]} $
$ = \frac{{{e^x}}}{{(1 - {e^x})\sqrt {1 - {e^{2x}}} }}$.
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$(A)$ $f^{\prime \prime}(x)$ exists for all $x \in(0, \infty)$
$(B)$ $f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$, but not differentiable on $(0, \infty)$
$(C)$ there exists $\alpha>1$ such that $\left|f^{\prime}(x)\right|<|f(x)|$ for all $x \in(\alpha, \infty)$
$(D)$ there exists $\beta>0$ such that $|f(x)|+\left|f^{\prime}(x)\right| \leq \beta$ for all $x \in(0, \infty)$