Find the indicated derivative: d/dx (x^(ln(x)/6))

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You need to use logarithmic diferentiation , hence, you should come up with the notation `y = (1/6)*x^(ln x)` .

You need to evaluate logarithms both sides such that:

`ln y = ln((1/6)*x^(ln x))`

Using the product property yields:

`ln y = ln(1/6) + ln x^(ln x)`

Using the power property of logarithms yields:

`ln y = ln(1/6) + ln^2 x`

You need to differentiate both sides with respect to x such that:

`(d(ln y))/(dx) = 0 + d((ln^2 x))/(dx) `

`(dy)/(ydx) = (2 ln x)/x`

You need to isolate (dy)/(dx) to the left side, hence you need to multiply by y both sides such that:

`(dy)/(dx) = (2 ln x) y/x`

You need to substitute `(1/6)*x^(lnx)`  for y such that:

`(dy)/(dx) = ((ln x^2)*x^(ln x))/(6x)`

Hence, evaluating `(dy)/(dx)`  using logarithmic differentiation yields `(dy)/(dx) = ((ln x^2)*x^(ln x))/(6x).`

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Let the function y =` x^(ln x)/6`

=> 6y = `x^(ln x)`

take the natural log of both the sides

`ln 6y = ln x*ln x`

use implicit differentiation

`(6/(6y))*(dy/dx) = 2*ln x*(1/x)`

=> `dy/dx = (2*y*ln x)/x`

=> `dy/dx = (2*x^(ln x)*ln x)/(6*x)`

The required derivative is `(x^(ln x)*ln x)/(3*x)`

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