Been learning calculus, and was taken aback (in a good way) by the challenge of
$$\frac{d}{dx}x^x$$
because the rules I've learned so far (chain/product/quotient/implicit differentiation) can handle $x^a$ and $a^x$ and many other functions, but not this special boy, which just takes 2 characters to write!
Now I've learned how you can solve it with logarithmic differentiation. But I'm curious.. is this a derivative you simply cannot solve without logarithmic differentiation? Could you use other derivative rules to solve this? If yes, which ones, and if no.. how do you identify equations that require logarithmic differentiation?
Note: I'm aware logarithmic differentiation utilizes implicit differentiation which I previously learned, but it also involves an extra step of logging both sides.
