Find the derivative of $2^{\tan(1/x)}$. I know that I should replace $\frac1x$ with $u$ and such, but then I can't continue it...
Derivatives question involving tangent
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derivatives
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0@RahulNarain Sorry, I was not aware that it would be problematic. I will scale down the editing. – 2012-11-22
1 Answers
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Hint: $2^{\tan(1/x)}=e^{(\ln 2)(\tan(1/x)}$.
If $y=e^u$, then by the Chain Rule, $\dfrac{dy}{dx}=e^u \dfrac{du}{dx}$.
Now let $u=(\ln 2)\tan(1/x)$. I think you know how to find $\dfrac{du}{dx}$.
Another way: Let $y=2^{\tan(1/x)}$. Take the natural logarithm of both side. We get $\ln y=(\ln 2)\tan(1/x).$ Now differentiate both sides with respect to $x$. On the left we get $\dfrac{1}{y}\dfrac{dy}{dx}$.
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0@Joe: Don't know. Small fractions render funny (with real LaTeX they look OK). – 2012-11-22