To find the derivative of \( y \) with respect to \( x \) in the given equation:
\[ -8 = 6y^3 - 9x^3 - 8x + 8x^2 \]
1. Start by rearranging the equation to isolate \( y \) on one side:
\[ 6y^3 = 9x^3 + 8x - 8x^2 - 8 \]
2. Divide by 6 to get \( y^3 \) on its own:
\[ y^3 = \frac{9x^3 + 8x - 8x^2 - 8}{6} \]
3. Now, take the cube root of both sides to solve for \( y \):
\[ y = \sqrt[3]{\frac{9x^3 + 8x - 8x^2 - 8}{6}} \]
4. To find the derivative of \( y \) with respect to \( x \), you need to apply the chain rule and the power rule for differentiation.
5. Differentiate the expression under the cube root with respect to \( x \) to find \( \frac{dy}{dx} \) or \( y' \), which represents the derivative of \( y \) with respect to \( x \).
6. Remember to simplify the derivative expression if possible.
By following these steps, you can find the derivative of \( y \) with respect to \( x \) in the given equation.
