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How can I calculate the derivative of the following function?

\begin{align*}(1+x)^{1/x}\end{align*}

It comes from here, If I use L'Hospital Rule to do the limit, then I should do the derivative, but I don't know how to do...

\begin{align*}\text{ }\lim_{x\to 0} \frac{e-(1+x)^{1/x}}{x}\end{align*}

Are there any other methods to do the limit?

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Write it as $\exp\left(\frac 1x \log(1 + x)\right)$, then use the chain rule and the product rule.

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Let $f(x)=(1+x)^{\frac{1}{x}}$, then $\log f(x)=\frac{1}{x}\log(1+x)$, thus $$\frac{f'(x)}{f(x)}=-\frac{1}{x^2}\log(1+x)+\frac{1}{x}\frac{1}{1+x}.$$

Michael

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Here you go:

$(1+x)^{1/x}$

Express $(x+1)^{1/x}$ as a power of e:

$(x+1)^{1/x} = e^{\frac{log((x+1)}{x}}$

Using the chain rule, and the product rule afterwards, gives you the following:

$\frac{(1+x)^{\frac{1}{x}}(\frac{x}{1+x}-log(1+x))}{x^2}$

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