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?
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$e^{ln(x)}$ is x, this is just the natural log raised to a number. – Set Jun 01 '15 at 01:41
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2@Tyroshipleasurebarge Look again at the question :) – zahbaz Jun 01 '15 at 01:44
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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}
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Thanks! What rule says I can't simplify that to just
xbefore using the power rule? – Jack Jun 02 '15 at 21:25 -
Because $\left(\ln x\right)^e\neq e^{\ln x}$, so it can't be simplified. – Eric R. Anschuetz Jun 02 '15 at 21:58