Hi,

I've been stumped trying to find the derivative $\displaystyle \frac{d}{dx} \left [x^{1/x} \right ]$.

By the chain, rule: $\displaystyle \frac{d}{dx} \left [x^{u} \right ]=ux^{u-1} \frac{du}{dx}$.

Substituting $\displaystyle u=\frac{1}{x}=x^{-1}$ and $\displaystyle \frac{du}{dx}=-x^{-2}$ results in: $\displaystyle \frac{d}{dx} \left [x^{1/x} \right ]=x^{-1} \cdot x^{\frac{1}{x}-1} \cdot -x^{-2}=-x^{\frac{1}{x}-4}$

However, that cannot be the correct derivative. By inspecting their graphs, $\displaystyle x^{1/x}$ appears to have a critical (turning) point, yet there is no solution to $\displaystyle 0=-x^{\frac{1}{x}-4}$.

Is there any reason for this apparent contradiction? Did I make a mistake?

Thanks in advance for any insight on this issue.