What is the derivative of
?
I was thinking of converting it to x^(1/6)(ln(x)), but after I take the derivative, my answer is always wrong.
$\displaystyle y = \left(\sqrt[6]{x}\right)^{\ln{x}}$
$\displaystyle = \left(x^{\frac{1}{6}}\right)^{\ln{x}}$
$\displaystyle = x^{\frac{1}{6}\ln{x}}$.
Therefore
$\displaystyle \ln{y} = \ln{\left(x^{\frac{1}{6}\ln{x}}\right)}$
$\displaystyle = \frac{1}{6}\ln{(x)}\ln{(x)}$
$\displaystyle = \frac{1}{6}\left(\ln{x}\right)^2$.
$\displaystyle \frac{d}{dx}(\ln{y}) = \frac{d}{dx}\left[\frac{1}{6}(\ln{x})^2\right]$
$\displaystyle \frac{d}{dy}(\ln{y})\,\frac{dy}{dx} = \frac{1}{6}\cdot \frac{1}{x}\cdot 2\ln{x}$
$\displaystyle \frac{1}{y}\,\frac{dy}{dx} = \frac{\ln{x}}{3x}$
$\displaystyle \frac{dy}{dx} = \frac{y\ln{x}}{3x}$
$\displaystyle \frac{dy}{dx} = \frac{\left(\sqrt[6]{x}\right)^{\ln{x}}\ln{x}}{3x}$.