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I need help understanding how to derivate this function:

$$f(x) = x^{\arctan(x)}$$.

Any suggestions?

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Is this homework? – Aryabhata Oct 11 '10 at 17:52
I am studying for an exam, this was from an old one and I am trying to understand it. – Mickel Oct 11 '10 at 17:54
Write it as g(h(x)) or maybe more steps, where each function is one you know how to differentiate. Apply the chain rule. – Ross Millikan Oct 11 '10 at 18:01
up vote 5 down vote accepted

Let $f(x) = x^{\arctan{x}}$ then $\log{f(x)} = \arctan{x} \cdot \log x$. Therefore $$\frac{1}{f(x)} \times f'(x) = \frac{d}{dx} \Bigl[ \arctan{x} \cdot \log x \Bigr] \Longrightarrow f'(x) = f(x) \times \frac{d}{dx} \Bigl[ \arctan{x} \cdot \log x \Bigr]$$

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HINT $\ \ g^{\:h}\ =\ e^{h\: \log(g)}\:.\ $ Or, take logs, cf. logarithmic derivative, and my post here.

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is there a name for that rule? Trying to find out more about it... – Mickel Oct 11 '10 at 18:10
exp and log are inverse functions: $\exp(\log\ g) = g$ on $\mathbb R^+$ – Bill Dubuque Oct 11 '10 at 18:33

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