Use implicit differentiation to determine \( \frac{d y}{d x} \) for the equation \( x^{2} \sqrt{y}-3 x^{5}-6=e^{2} \)

To use implicit differentiation on the equation \( x^{2} \sqrt{y} - 3 x^{5} - 6 = e^{2} \), we can start by differentiating both sides with respect to \( x \). 1. Calculate the derivative of the left side: \[ \frac{d}{dx}(x^{2} \sqrt{y}) = x^{2} \cdot \frac{d}{dx}(\sqrt{y}) + \sqrt{y} \cdot \frac{d}{dx}(x^{2}) = x^{2} \cdot \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} + \sqrt{y} \cdot 2x \] 2. Now derive \( -3x^5 \) and \( -6 \): \[ \frac{d}{dx}(-3x^{5}) = -15x^{4} \] \[ \frac{d}{dx}(-6) = 0 \] 3. The right side, \( e^{2} \), is a constant, so its derivative is: \[ \frac{d}{dx}(e^{2}) = 0 \] Now, putting everything together, we differentiate the whole equation: \[ x^{2} \cdot \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} + 2x\sqrt{y} - 15x^{4} = 0 \] 4. Rearranging gives: \[ x^{2} \cdot \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 15x^{4} - 2x\sqrt{y} \] 5. To isolate \( \frac{dy}{dx} \), multiply both sides by \( \frac{2\sqrt{y}}{x^{2}} \): \[ \frac{dy}{dx} = \frac{2\sqrt{y}(15x^{4} - 2x\sqrt{y})}{x^{2}} \] 6. Thus, we simplify further: \[ \frac{dy}{dx} = \frac{2\sqrt{y}(15x^{4} - 2x\sqrt{y})}{x^{2}} \] This gives the final result for \( \frac{dy}{dx} \).