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Câu hỏi: Cho hàm số $y=\dfrac{\ln x}{x}.$ Mệnh đề nào dưới đây đúng?
A. $2{y}'+x{{y}'}'=-\dfrac{1}{{{x}^{2}}}.$
B. ${y}'+x{{y}'}'=\dfrac{1}{{{x}^{2}}}.$
C. ${y}'+x{{y}'}'=-\dfrac{1}{{{x}^{2}}}.$
D. $2{y}'+x{{y}'}'=\dfrac{1}{{{x}^{2}}}.$
A. $2{y}'+x{{y}'}'=-\dfrac{1}{{{x}^{2}}}.$
B. ${y}'+x{{y}'}'=\dfrac{1}{{{x}^{2}}}.$
C. ${y}'+x{{y}'}'=-\dfrac{1}{{{x}^{2}}}.$
D. $2{y}'+x{{y}'}'=\dfrac{1}{{{x}^{2}}}.$
${y}'=\dfrac{\dfrac{1}{x}.x-\ln x}{{{x}^{2}}}=\dfrac{1-\ln x}{{{x}^{2}}}; {{y}'}'=\dfrac{-\dfrac{1}{x}.{{x}^{2}}-2x (1-\ln x)}{{{x}^{4}}}=\dfrac{-3x+2x\ln x}{{{x}^{4}}}=\dfrac{-3+2\ln x}{{{x}^{3}}}$
$\Rightarrow x{{y}'}'+2{y}'=\dfrac{-3+2\ln x+2-2\ln x}{{{x}^{2}}}=-\dfrac{1}{{{x}^{2}}}.$
$\Rightarrow x{{y}'}'+2{y}'=\dfrac{-3+2\ln x+2-2\ln x}{{{x}^{2}}}=-\dfrac{1}{{{x}^{2}}}.$
Đáp án A.