Y=x^1/x⇒ log y =log x/x⇒y_1/y =1 log x/x^2 = y.1 log x/x^2 = x^1/x.1 log x/x^2 ...

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Derivative of Standard Functions

d/dx x1/x=

Question

$\frac{d}{d x} {x^{1 / x}} =$

$x^{1 / x - 2} (1 + l o g x)$

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$x^{\frac{1}{x} - 2} (1 - l o g x)$

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1+logx

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$1 + l o g x^{1 / x - 2}$

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Solution

The correct option is B
$x^{\frac{1}{x} - 2} (1 - l o g x)$

$y = x^{\frac{1}{x}} \Rightarrow l o g y$
$= \frac{l o g x}{x} \Rightarrow \frac{y_{1}}{y}$
$= \frac{1 - l o g x}{x^{2}}$
$= y . \frac{1 - l o g x}{x^{2}}$
$= x^{1 / x} . \frac{1 - l o g x}{x^{2}}$

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