What is the derivative of $x^((1/5)(lnx))$ ?

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Answer 1 · 2016-01-07T09:26:24

$2/5*x^((1/5*ln x-1))* ln x$

Let $y = x^((1/5*ln x))$ then take logarithm of both sides of the equation

$ln y = ln x^((1/5*ln x))$

$ln y = (1/5 ln x )*( ln x)$

$ln y = 1/5*(ln x)^2$

$1/y * y' = 1/5* 2* ln x*1/x$

then solve for $y'$

$y' = (2y)/(5x) ln x$

replace $y$ with the equivalent $x^((1/5*ln x))$

$y' = (2x^((1/5*ln x)))/(5x) ln x$

then simplification

$y' = 2/5*x^((1/5*ln x-1))* ln x$

What is the derivative of x^((1/5)(lnx))x^((1/5)(lnx))?