A) \[\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right)\centerdot {{7}^{x+\frac{1}{x}}}\centerdot \log 7\]
B) \[\left( \frac{{{x}^{2}}+1}{{{x}^{2}}} \right)\centerdot {{7}^{x+\frac{1}{x}}}\centerdot \log 7\]
C) \[\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right)\centerdot {{7}^{x-\frac{1}{x}}}\centerdot \log 7\]
D) \[\left( \frac{{{x}^{2}}+1}{{{x}^{2}}} \right)\centerdot {{7}^{x+\frac{1}{x}}}\centerdot \log 7\]
Correct Answer: A
Solution :
Let \[y={{7}^{x+\frac{1}{x}}}\] \[\therefore \,\frac{dy}{dx}=\frac{d}{dx}\left( {{7}^{x+\frac{1}{x}}} \right)\] \[={{7}^{x+\frac{1}{x}}}.\,\log \,7.\,\frac{d}{dx}\left( x+\frac{1}{x} \right)={{7}^{x+\frac{1}{x}}}.\log \,7.\left( 1-\frac{1}{{{x}^{2}}} \right)\] \[=\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right){{.7}^{x+\frac{1}{x}}}.\log \,7\]
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