How do you differentiate $(1+x)^(1/x)$ ?

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Answer 1 · 2017-02-16T12:45:41

$f'(x) = (1+x)^(1/x) ( (x-(1+x)ln(1+x))/(x^2(1+x)))$

The function:

$f(x) = (1+x)^(1/x)$

has only positive values, so we can take its logarithm:

$ln f(x) = ln((1+x)^(1/x)) = ln(1+x)/x$

Differentiate now the equation above:

$d/dx ln (f(x)) = d/dx (ln(1+x)/x)$

$(f'(x))/f(x) =( (x/(1+x))-ln(1+x))/x^2$

$f'(x) = f(x) ( (x-(1+x)ln(1+x))/(x^2(1+x)))$

$f'(x) = (1+x)^(1/x) ( (x-(1+x)ln(1+x))/(x^2(1+x)))$

How do you differentiate (1+x)^(1/x)(1+x)^(1/x)?