1. ## Exponential Derivative

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.

2. Originally Posted by Kimmy2
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.
$y = \left(\sqrt[6]{x}\right)^{\ln{x}}$

$= \left(x^{\frac{1}{6}}\right)^{\ln{x}}$

$= x^{\frac{1}{6}\ln{x}}$.

Therefore

$\ln{y} = \ln{\left(x^{\frac{1}{6}\ln{x}}\right)}$

$= \frac{1}{6}\ln{(x)}\ln{(x)}$

$= \frac{1}{6}\left(\ln{x}\right)^2$.

$\frac{d}{dx}(\ln{y}) = \frac{d}{dx}\left[\frac{1}{6}(\ln{x})^2\right]$

$\frac{d}{dy}(\ln{y})\,\frac{dy}{dx} = \frac{1}{6}\cdot \frac{1}{x}\cdot 2\ln{x}$

$\frac{1}{y}\,\frac{dy}{dx} = \frac{\ln{x}}{3x}$

$\frac{dy}{dx} = \frac{y\ln{x}}{3x}$

$\frac{dy}{dx} = \frac{\left(\sqrt[6]{x}\right)^{\ln{x}}\ln{x}}{3x}$.

3. I see. I did not differentiate at all. I ended up doing something like this:

(1/6)lnx^(lnx)

Then

lnx * lnx * 1/6 * lnx

Which was completely wrong.
Thank you so much, your explanation was so clear. I understand!