use implicit differentiation to find dx/dy.

Question

x^4y^2 -x^3y +2xy^3=0

Answer 1

The function is x⁴y² - x³y + 2xy³ = 0.

Differentiate the function with respect to y.

(x⁴y²) ' - (x³y) ' + (2xy³) ' = 0

Apply product rule : (uv) ' = uv ' + vu '.

x⁴(y²) ' + y²(x⁴) ' - x³(y) ' - y(x³) ' + 2x(y³) ' + 2(y³)x ' = 0

Apply power rule : (xⁿ) ' = nx^{n - 1} .

x⁴(2y) + y²(4x³) x ' - x³(1) - y(3x²) x ' + 2x(3y²) + 2(y³)x ' = 0

2x⁴y + 4x³y²x ' - x³ - 3x²yx ' + 6xy² + 2y³x ' = 0

4x³y²x ' - 3x²yx ' + 2y³x ' = x³ - 6xy² - 2x⁴y

x ' = (x³ - 6xy² - 2x⁴y) / (4x³y² - 3x²y + 2y³).

Therefore, dx / dy = (x³ - 6xy² - 2x⁴y) / (4x³y² - 3x²y + 2y³).