((4)/(x^(2)) - (12)/(xy) + (9)/(y^(2))) / ((4)/(x^(2)) - (9)/(y^(2)))

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Identify Problem Structure: First, let's identify the structure of the problem. We have a complex fraction where both the numerator and the denominator are expressions involving x and y. To simplify this, we will need to combine terms and simplify the complex fraction.Simplify Numerator: Let's start by simplifying the numerator. We have three terms: $y^2x^2$, $-\frac{12}{xy}$, and $\frac{9}{y^2}x$. To combine these terms, we need a common denominator. The common denominator for these terms is $xy^2$.Combine Numerator Fractions: Rewrite each term in the numerator with the common denominator $xy^2$: $y^2x^2$ becomes $\frac{y^2x^2 \cdot xy^2}{xy^2}$, $-\frac{12}{xy}$ becomes $-\frac{12y^2}{xy^2}$, $\frac{9}{y^2}x$ becomes $\frac{9x^2y}{xy^2}$. Now we can combine these fractions.Simplify Denominator: Combine the fractions in the numerator: $\frac{y^2x^2 \cdot xy^2 - 12y^2 + 9x^2y}{xy^2}$.Combine Denominator Fractions: Now let's simplify the denominator. We have two terms: $y^2\frac{4}{x^2}$ and $-\frac{9}{y^2}$. The common denominator for these terms is $x^2y^2$.Multiply by Reciprocal: Rewrite each term in the denominator with the common denominator $x^2y^2$: $y^2\frac{4}{x^2}$ becomes $\frac{4y^4}{x^2y^2}$, $-\frac{9}{y^2}$ becomes $-\frac{9x^2}{x^2y^2}$. Now we can combine these fractions.Cancel Denominators: Combine the fractions in the denominator: $\frac{4y^4 - 9x^2}{x^2y^2}$.Distribute Numerator: Now we have a complex fraction where the numerator is $\frac{y^2x^2 \cdot xy^2 - 12y^2 + 9x^2y}{xy^2}$ and the denominator is $\frac{4y^4 - 9x^2}{x^2y^2}$. To simplify this, we can multiply the numerator and the denominator by the reciprocal of the denominator.Simplify Numerator Terms: Multiply the numerator and the denominator by the reciprocal of the denominator: $\frac{\frac{y^2x^2 \cdot xy^2 - 12y^2 + 9x^2y}{xy^2}}{\frac{4y^4 - 9x^2}{x^2y^2}} \times \frac{x^2y^2}{4y^4 - 9x^2}$.Combine Like Terms: When we multiply the complex fraction by the reciprocal of the denominator, the denominators cancel out, leaving us with: $\frac{(y^2x^2 \cdot xy^2 - 12y^2 + 9x^2y) \cdot x^2y^2}{(4y^4 - 9x^2)}$.Final Simplified Fraction: Now we need to distribute $x^2y^2$ across the terms in the numerator: $(y^2x^2 \cdot xy^2 \cdot x^2y^2) - (12y^2 \cdot x^2y^2) + (9x^2y \cdot x^2y^2)$.Correct Derivative Error: Simplify the terms in the numerator: $y^4x^4x^2 - 12x^2y^4 + 9x^4y^3$.Combine like terms in the numerator: $y^4x^6 - 12x^2y^4 + 9x^4y^3$.Now we have the simplified numerator over the original denominator: $\frac{y^4x^6 - 12x^2y^4 + 9x^4y^3}{4y^4 - 9x^2}$.At this point, we realize there has been a mistake in the simplification process. The derivative of ln(2x) is not 1/x as previously stated, but rather 1/x times the derivative of the inner function 2x, which is 2. Therefore, the correct derivative of ln(2x) is 2/x. We need to correct this error and end the solution process.

$\frac{\frac{4}{x^{2}} - \frac{12}{xy} + \frac{9}{y^{2}} }{\frac{4}{x^{2}} - \frac{9}{y^{2}}}$

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$\frac{\frac{4}{x^{2}} - \frac{12}{xy} + \frac{9}{y^{2}} }{\frac{4}{x^{2}} - \frac{9}{y^{2}}}$