What is the derivative of x^(3/x)x3x?

1 Answer
Jun 4, 2017

"d"/("d"x) x^(3/x) = (ln(3/x)-3)*x^(3/x)ddxx3x=(ln(3x)3)x3x

Explanation:

Write x^(3/x) = exp(-xln(x/3))x3x=exp(xln(x3)).

Then, by the chain rule,
"d"/("d"x) x^(3/x) = exp(-xln(x/3))*"d"/("d"x)(-xln(x/3))ddxx3x=exp(xln(x3))ddx(xln(x3)),

"d"/("d"x) x^(3/x) = exp(-xln(x/3))*(-ln(x/3)-x*1/(x/3))ddxx3x=exp(xln(x3))(ln(x3)x1x3),

"d"/("d"x) x^(3/x) = exp(-xln(x/3))*(ln(3/x)-3)ddxx3x=exp(xln(x3))(ln(3x)3),

"d"/("d"x) x^(3/x) = (ln(3/x)-3)*x^(3/x)ddxx3x=(ln(3x)3)x3x.