Question

Consider y as the independent variable and x as the dependent variable. Use implicit differentiation to find dx/dy for the equation x³y² - x³y - 5xy³ = 0.

Answer 1

To find dx/dy for the equation x³y² - x³y - 5xy³ = 0 using implicit differentiation, take the derivative of each term with respect to y, apply the chain rule for terms with x, simplify, and solve for dx/dy.

Step-by-step explanation:

To find dx/dy using implicit differentiation for the equation x³y² - x³y - 5xy³ = 0, we take the derivative of both sides of the equation with respect to y. Remember that since we are differentiating with respect to y, and x is considered the dependent variable, every time we differentiate a term with x, we must multiply by dx/dy to account for the chain rule.

Following these steps, we have:

1. Derivative of x³y²: 2x³y (dy/dy) + y² (3x²)(dx/dy)
2. Derivative of -x³y: -x³ (dy/dy) - y (3x²)(dx/dy)
3. Derivative of -5xy³: -5x (3y²)(dy/dy) - 5y³ (dx/dy)

Now, we simplify and solve for dx/dy:

2x³y - x³ - 15xy² + (y² - 3xy + 15y³) dx/dy = 0

Isolate the dx/dy terms:

(y² - 3xy + 15y³) dx/dy = 3x³ - 2x³y + 15xy²

Finally:

dx/dy = (3x³ - 2x³y + 15xy²) / (y² - 3xy + 15y³)