Please help, thanks! Consider x^2y−y^3=6.

Question

Compute dy/dx.?

Answer 1

x²y - y³ = 6

Apply 'derivative with respect to x ' each side.

(d/dx)(x²y - y³) = (d/dx)6

(d/dx)(x²y) - (d/dx)(y³) = (d/dx)6

Derivative formulas: (d/dx)constant = 0, (d/dx)(xⁿ) = nx^n-1, and (UV)'=U'V + UV'

y(2x) + x²y' - 3y²y' = 0

Subtract 2xy from each side.

x²y' - 3y²y' = - 2xy

Take out common term y'.

y'(x² - 3y²) = - 2xy.

Divide each side by (x² - 3y²).

So, y' = (-2xy)/(x² - 3y²)

Therefore dy/dx = (-2xy) / (x² - 3y²).