Find \( \frac{d y}{d x} \) for the equation \( x^{9} y^{2}=2 \) using implicit differentiation.

To find \( \frac{d y}{d x} \) using implicit differentiation for the equation \( x^{9} y^{2} = 2 \), we start by differentiating both sides of the equation with respect to \( x \). 1. Differentiate the left side using the product rule: \[ \frac{d}{dx}(x^{9} y^{2}) = \frac{d}{dx}(x^{9}) \cdot y^{2} + x^{9} \cdot \frac{d}{dx}(y^{2}). \] The derivative of \( x^{9} \) is \( 9x^{8} \), and for \( y^{2} \), we apply the chain rule: \[ \frac{d}{dx}(y^{2}) = 2y \frac{dy}{dx}. \] Thus, we have: \[ 9x^{8} y^{2} + x^{9} \cdot 2y \frac{dy}{dx}. \] 2. The derivative of the right side \( \frac{d}{dx}(2) \) is \( 0 \). Now we equate the derivatives: \[ 9x^{8} y^{2} + x^{9} \cdot 2y \frac{dy}{dx} = 0. \] 3. Next, we isolate \( \frac{dy}{dx} \): \[ x^{9} \cdot 2y \frac{dy}{dx} = -9x^{8} y^{2}. \] \[ \frac{dy}{dx} = -\frac{9x^{8} y^{2}}{2x^{9} y}. \] 4. Simplifying this gives: \[ \frac{dy}{dx} = -\frac{9y}{2x}. \] Therefore, the final result is: \[ \frac{dy}{dx} = -\frac{9y}{2x}. \]