What's in the denominator?

Calculus Level 3

Given that for \(x>0\), \(y=y(x)>0\), \[e^{-\cot(y)}=x^x+x\ln(x)\]

and implicit differentiation of the above equation gives \[\frac{dy}{dx}=\frac{(x^x+1)(\ln(x)+1)}{(\csc^2(y))(x^x+xf(x))}\]

where \(f(x)\) is a function involving \(x\) (not necessarily equalling \(y\)), find \(f(x)\).

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