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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?2010-10-11
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    Write it as g(h(x)) or maybe more steps, where each function is one you know how to differentiate. Apply the chain rule.2010-10-11

2 Answers 2

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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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