Let $\displaystyle \alpha > 0$. Prove that no matter how small $\displaystyle \alpha$ is, there is an $\displaystyle N \epsilon$Rsuch that $\displaystyle ln x \leq x^{\alpha}$ for all $\displaystyle x \leq N$. Note: N depends on $\displaystyle \alpha$.

I need to use the mean value theorem to do this.

I started by finding the difference of the derivatives of both functions.

Then didnt know where to go... Any help would be greatly appreciated...