What is the derivative of $x^(3/x)$ ?

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Answer 1 · 2017-06-04T21:06:05

$"d"/("d"x) x^(3/x) = (ln(3/x)-3)*x^(3/x)$

Write $x^(3/x) = exp(-xln(x/3))$ .

Then, by the chain rule,
$"d"/("d"x) x^(3/x) = exp(-xln(x/3))*"d"/("d"x)(-xln(x/3))$ ,

$"d"/("d"x) x^(3/x) = exp(-xln(x/3))*(-ln(x/3)-x*1/(x/3))$ ,

$"d"/("d"x) x^(3/x) = exp(-xln(x/3))*(ln(3/x)-3)$ ,

$"d"/("d"x) x^(3/x) = (ln(3/x)-3)*x^(3/x)$ .

What is the derivative of x^(3/x)x^(3/x)?