Find the derivative of the composite algebraic-trigonometric function expression with respect to $x$:
\[
f(x) = x^{\cot x} + \frac{2x^2 - 3}{2x^2 - x + 2}
\]
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When applying the quotient rule, be extra careful with negative signs when expanding polynomial terms in the numerator. A single sign error during expansion will change the final quadratic coefficients!