Find \( y^{\prime} \) if \( y=\ln \left(9 x^{2}+5 y^{2}\right) \) \( y^{\prime}=\square \)

Find the first order derivative with respect to \( x \) for \( y=\ln(9x^2+5y^2) \). Find the first derivative by following steps: - step0: Find the derivative with respect to \(x\): \(y=\ln{\left(9x^{2}+5y^{2}\right)}\) - step1: Take the derivative: \(\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(\ln{\left(9x^{2}+5y^{2}\right)}\right)\) - step2: Calculate the derivative: \(\frac{dy}{dx}=\frac{d}{dx}\left(\ln{\left(9x^{2}+5y^{2}\right)}\right)\) - step3: Calculate the derivative: \(\frac{dy}{dx}=\frac{18x+10y\frac{dy}{dx}}{9x^{2}+5y^{2}}\) - step4: Cross multiply: \(\frac{dy}{dx}\times \left(9x^{2}+5y^{2}\right)=18x+10y\frac{dy}{dx}\) - step5: Simplify the equation: \(\left(9x^{2}+5y^{2}\right)\frac{dy}{dx}=18x+10y\frac{dy}{dx}\) - step6: Move the variable to the left side: \(\left(9x^{2}+5y^{2}\right)\frac{dy}{dx}-10y\frac{dy}{dx}=18x\) - step7: Collect like terms: \(\left(9x^{2}+5y^{2}-10y\right)\frac{dy}{dx}=18x\) - step8: Divide both sides: \(\frac{\left(9x^{2}+5y^{2}-10y\right)\frac{dy}{dx}}{9x^{2}+5y^{2}-10y}=\frac{18x}{9x^{2}+5y^{2}-10y}\) - step9: Divide the numbers: \(\frac{dy}{dx}=\frac{18x}{9x^{2}+5y^{2}-10y}\) The derivative of \( y \) with respect to \( x \) is given by: \[ y' = \frac{18x}{9x^{2}+5y^{2}-10y} \] Therefore, the derivative of \( y \) with respect to \( x \) is \( \frac{18x}{9x^{2}+5y^{2}-10y} \).