A) \[\frac{{{(-1)}^{n-1}}(n-1)!}{{{x}^{n}}}\]
B) \[\frac{{{(-1)}^{n}}(n-1)!}{{{x}^{n-1}}}\]
C) \[\frac{{{(-1)}^{n}}n!}{{{x}^{n-1}}}\]
D) \[\frac{{{(-1)}^{n}}(n-1)!}{{{x}^{n-1}}}\]
Correct Answer: A
Solution :
Let \[y=log\text{ }x\] \[\frac{dy}{dx}=\frac{1}{x}\] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=-\frac{1}{{{x}^{2}}}={{(-1)}^{1}}\frac{1!}{{{x}^{2}}}\] \[\frac{{{d}^{3}}y}{d{{x}^{3}}}=\frac{2}{{{x}^{2}}}={{(-1)}^{2}}\frac{2!}{{{x}^{3}}}\] \[\frac{{{d}^{4}}y}{d{{x}^{4}}}=\frac{-2.3}{{{x}^{4}}}={{(-1)}^{3}}\frac{3!}{{{x}^{4}}}\] ????????????. ????????????. ????????????. Hence, \[\frac{{{d}^{n}}y}{d{{x}^{n}}}={{(-1)}^{n-1}}\frac{(n-1)!}{{{x}^{n}}}\]
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