A) \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{2x}}}}\]
B) \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{x}}}}\]
C) \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1+{{e}^{2x}}}}\]
D) \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1+{{e}^{x}}}}\]
Correct Answer: A
Solution :
\[y=\sqrt{\frac{1+{{e}^{x}}}{1-{{e}^{x}}}}\]or \[{{y}^{2}}=\frac{1+{{e}^{x}}}{1-{{e}^{x}}}\] \[2y\frac{dy}{dx}=\frac{(1-{{e}^{x}}){{e}^{x}}+(1+{{e}^{x}}){{e}^{x}}}{{{(1-{{e}^{x}})}^{2}}}\] \[=\frac{2{{e}^{x}}}{{{(1-{{e}^{x}})}^{2}}}\] \[\therefore \] \[\frac{dy}{dx}=\frac{{{e}^{x}}}{{{(1-{{e}^{x}})}^{2}}}\sqrt{\left( \frac{1-{{e}^{x}}}{1+{{e}^{x}}} \right)\left( \frac{1-{{e}^{x}}}{1-{{e}^{x}}} \right)}\] \[=\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{2x}}}}\]
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