A) 1
B) \[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\]
C) \[\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\]
D) \[\frac{{{x}^{2}}-1}{{{x}^{2}}-4}{{e}^{x}}\]
Correct Answer: C
Solution :
\[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x-2}{x+2} \right)}^{3/4}} \right\} \right]\] Let \[y=\log \left\{ {{e}^{x}}{{\left( \frac{x-2}{x+2} \right)}^{3/4}} \right\}\] \[=\log {{e}^{x}}+\frac{3}{4}\log \left( \frac{x-2}{x+2} \right)\] \[=x+\frac{3}{4}+\log (x-2)-\frac{3}{4}\log (x+2)\] Differentiating w.r.t. \['x'\] \[\frac{dy}{dx}=1+\frac{3}{4(x-2)}-\frac{3}{4(x+2)}\] \[=\frac{4(x-2)(x+2)+3(x+2)-3(x-2)}{4(x-2)(x+2)}\] \[=\frac{4{{x}^{2}}-16+3x+6-3x+6}{4({{x}^{4}}-4)}\] \[=\frac{4{{x}^{2}}-4}{4({{x}^{2}}-4)}\] \[=\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\]
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