Question: Based on a question of Peter Danenhower, and comments by Alan Cooper.

Write down the power rule: \(\frac{d}{d x}\left(x^{n}\right)=\) And the exponential rule: \(\frac{d}{d x}\left(a^{x}\right)=\) Now what do you get if you use the power rule to find \(\frac{d}{d x}\left(x^{x}\right)\) ? What do you get if you use the exponential rule to find \(\frac{d}{d x}\left(x^{x}\right) ?\) Both are wrong of course, \(x^{x}\) is neither a power nor an exponential function, it's kind of both!

So add the two results that you obtained: \(\frac{d}{d x}\left(x^{x}\right)=\) Is the answer correct now? YES / NO (it should be, but why? adding them isn't logical of itself)

Consider now the function $z=y^{x} \quad(y>0)$. (This used to be on calculators, but it's largely been replaced by "^", called a "caret", or up arrow, except for when we call it "hat!")

Write down \(\frac{\partial z}{\partial x}=\_\_\_\_\_\_\_\frac{\partial z}{\partial y}=\_\_\_\_\_\_\_\_\_\_\)

Now consider the composite function:

Use the chain rule to find \(\frac{d z}{d x}\) at \((x, x)\) (you will sub \(y=x\) into your answer)

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