Use impleit differentiation to determine \( \frac{d y}{d x} \) ror the equation \( x^{4} \sqrt{y}-9 x^{6}-6-c^{4} \)

To use implicit differentiation on the equation \( x^{4} \sqrt{y} - 9 x^{6} - 6 - c^{4} = 0 \), we first differentiate both sides with respect to \( x \): 1. Differentiate \( x^{4} \sqrt{y} \) using the product rule: \[ \frac{d}{dx}(x^{4} \sqrt{y}) = 4x^{3} \sqrt{y} + x^{4} \cdot \frac{1}{2\sqrt{y}} \frac{dy}{dx} \] 2. The derivative of \( -9 x^{6} \) is \( -54 x^{5} \). 3. The derivatives of constants (like \(-6\) and \(-c^{4}\)) are \(0\). Putting it all together: \[ 4x^{3} \sqrt{y} + \frac{x^{4}}{2\sqrt{y}} \frac{dy}{dx} - 54 x^{5} = 0 \] Next, isolate \( \frac{dy}{dx} \): \[ \frac{x^{4}}{2\sqrt{y}} \frac{dy}{dx} = 54 x^{5} - 4x^{3} \sqrt{y} \] \[ \frac{dy}{dx} = \frac{(54 x^{5} - 4x^{3} \sqrt{y}) \cdot 2\sqrt{y}}{x^{4}} \] \[ = \frac{108 x^{5} \sqrt{y} - 8 x^{3} y}{x^{4}} \] \[ = 108 \frac{x\sqrt{y}}{1} - 8 \frac{y}{x} \] Therefore, \( \frac{dy}{dx} \) simplifies to: \[ \frac{dy}{dx} = 108 x \sqrt{y} - \frac{8y}{x} \] And that's the derivative!