Derivative of $(\ln x)^e$

Question

In Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result "looks like it should be $1$ or something, but it's not." Why is this? And what's the actual answer?

Answer 1

By the chain rule: \begin{equation} \begin{aligned} \frac{\textrm{d}}{\textrm{d}x}\left[\ln\left(x\right)\right]^e&=\frac{e\left[\ln\left(x\right)\right]^{e-1}}{x} \end{aligned} \end{equation}