Find \( \frac{d y}{d x} \) by implicit differentiation \[ \begin{array}{l}2 \sqrt{x}-9 \sqrt{y}=6\end{array} \]

To find \( \frac{dy}{dx} \) by implicit differentiation for the equation \( 2 \sqrt{x} - 9 \sqrt{y} = 6 \), we begin by differentiating both sides with respect to \( x \): 1. Differentiate \( 2 \sqrt{x} \): \[ \frac{d}{dx} (2 \sqrt{x}) = 2 \cdot \frac{1}{2} x^{-1/2} = \frac{1}{\sqrt{x}} \] 2. Differentiate \( -9 \sqrt{y} \): \[ \frac{d}{dx} (-9 \sqrt{y}) = -9 \cdot \frac{1}{2} y^{-1/2} \frac{dy}{dx} = -\frac{9}{2\sqrt{y}} \frac{dy}{dx} \] 3. Differentiate the constant \( 6 \): \[ \frac{d}{dx} (6) = 0 \] Now we can combine these derivatives in the original equation: \[ \frac{1}{\sqrt{x}} - \frac{9}{2\sqrt{y}} \frac{dy}{dx} = 0 \] Next, isolate \( \frac{dy}{dx} \): \[ -\frac{9}{2\sqrt{y}} \frac{dy}{dx} = -\frac{1}{\sqrt{x}} \] \[ \frac{dy}{dx} = \frac{1}{\sqrt{x}} \cdot \frac{2\sqrt{y}}{9} = \frac{2\sqrt{y}}{9\sqrt{x}} \] Thus, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = \frac{2\sqrt{y}}{9\sqrt{x}} \]