What is the derivative of $x^{\frac{3}{x}}$ $x^(3/x)$ ?

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What is the derivative of $x^{\frac{3}{x}}$ $x^(3/x)$ ?

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Answer 1 · 2017-06-04T21:06:05

$\frac{d}{d x} x^{\frac{3}{x}} = (ln (\frac{3}{x}) - 3) \cdot x^{\frac{3}{x}}$ $"d"/("d"x) x^(3/x) = (ln(3/x)-3)*x^(3/x)$

Explanation:

Write $x^{\frac{3}{x}} = exp (- x ln (\frac{x}{3}))$ $x^(3/x) = exp(-xln(x/3))$ .

Then, by the chain rule,
$\frac{d}{d x} x^{\frac{3}{x}} = exp (- x ln (\frac{x}{3})) \cdot \frac{d}{d x} (- x ln (\frac{x}{3}))$ $"d"/("d"x) x^(3/x) = exp(-xln(x/3))*"d"/("d"x)(-xln(x/3))$ ,

$\frac{d}{d x} x^{\frac{3}{x}} = exp (- x ln (\frac{x}{3})) \cdot (- ln (\frac{x}{3}) - x \cdot \frac{1}{\frac{x}{3}})$ $"d"/("d"x) x^(3/x) = exp(-xln(x/3))*(-ln(x/3)-x*1/(x/3))$ ,

$\frac{d}{d x} x^{\frac{3}{x}} = exp (- x ln (\frac{x}{3})) \cdot (ln (\frac{3}{x}) - 3)$ $"d"/("d"x) x^(3/x) = exp(-xln(x/3))*(ln(3/x)-3)$ ,

$\frac{d}{d x} x^{\frac{3}{x}} = (ln (\frac{3}{x}) - 3) \cdot x^{\frac{3}{x}}$ $"d"/("d"x) x^(3/x) = (ln(3/x)-3)*x^(3/x)$ .

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