Use implicit differentiation to find \( \frac{d y}{d x} \) \[ x^{3}=\frac{9 x-y}{x+5 y} \]

We start with the given equation:   x³ = (9x – y) / (x + 5y) Step 1. Eliminate the fraction by cross multiplying:   x³(x + 5y) = 9x – y Step 2. Expand the left-hand side:   x⁴ + 5x³y = 9x – y Step 3. Differentiate both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating any term with y, use the chain rule. Differentiate the left-hand side term by term: • d/dx (x⁴) = 4x³ • d/dx (5x³y): Use the product rule, where u = 5x³ and v = y.   u' = 15x² and v' = dy/dx,   so d/dx (5x³y) = (5x³)(dy/dx) + (15x²)(y) = 5x³ (dy/dx) + 15x²y Thus, the derivative of the left-hand side becomes:   4x³ + 15x²y + 5x³ (dy/dx) Now differentiate the right-hand side: • d/dx (9x) = 9 • d/dx (–y) = –(dy/dx) So the derivative of the right-hand side is:   9 – (dy/dx) Step 4. Set the derivatives equal to each other:   4x³ + 15x²y + 5x³ (dy/dx) = 9 – (dy/dx) Step 5. Collect all terms involving dy/dx on one side of the equation. Add (dy/dx) to both sides to get:   5x³ (dy/dx) + (dy/dx) = 9 – 4x³ – 15x²y Factor dy/dx from the left-hand side:   (dy/dx)(5x³ + 1) = 9 – 4x³ – 15x²y Step 6. Finally, solve for dy/dx by dividing both sides by (5x³ + 1):   dy/dx = (9 – 4x³ – 15x²y) / (5x³ + 1) This is the derivative of y with respect to x obtained by implicit differentiation.