How do you differentiate $x^{\frac{1}{x}}$ $x^(1/x)$ ?

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How do you differentiate $x^{\frac{1}{x}}$ $x^(1/x)$ ?

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Answer 1 · 2016-10-30T16:59:50

Use some version of logarithmic differentiation.

Explanation:

$x^{\frac{1}{x}} = e^{\frac{1}{x} ln x} = e^{\frac{ln x}{x}}$ $x^(1/x) = e^(1/xlnx) = e^(lnx/x)$

$\frac{d}{d x} (x^{\frac{1}{x}}) = \frac{d}{d x} (e^{\frac{ln x}{x}}) = e^{\frac{ln x}{x}} \cdot \frac{d}{d x} (\frac{ln x}{x})$ $d/dx(x^(1/x)) = d/dx (e^(lnx/x)) = e^(lnx/x) * d/dx(lnx/x)$

$= x^{\frac{1}{x}} \cdot \frac{(\frac{1}{x}) \cdot (x) - (ln x) (1)}{x^{2}}$ $= x^(1/x) * ((1/x)*(x) - (lnx)(1))/x^2$

$= x^{\frac{1}{x}} (\frac{1 - ln x}{x^{2}})$ $= x^(1/x)((1-lnx)/x^2)$

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How do you differentiate $x^{\frac{1}{x}}$ $x^(1/x)$ ?

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