# What is the derivative of #f(x)=x/ln(1/x)#?

And now we can substitute back into the original equation to obtain

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To find the derivative of ( f(x) = \frac{x}{\ln(1/x)} ), we can use the quotient rule:

[ f'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2} ]

Where ( f(x) = x ) and ( g(x) = \ln(1/x) ).

[ f'(x) = \frac{1 \cdot \ln(1/x) - x \cdot \frac{d}{dx}(\ln(1/x))}{(\ln(1/x))^2} ]

Using the chain rule and the derivative of ( \ln(x) ), we get:

[ f'(x) = \frac{\ln(1/x) + x \cdot \frac{1}{1/x} \cdot (-1/x^2)}{(\ln(1/x))^2} ]

[ f'(x) = \frac{\ln(1/x) - x \cdot (-x)}{x^2(\ln(1/x))^2} ]

[ f'(x) = \frac{\ln(1/x) + x^2}{x^2(\ln(1/x))^2} ]

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