Given
sin(xy) + x² + y³ = c
differentiating both sides with respect to x would give
d/dx [sin(xy)] + d/dx[x²] + d/dx[y³] = 0
By the chain and power rules,
cos(xy) d/dx[xy] + 2x + 3y² dy/dx = 0
By the product rule,
cos(xy) (x dy/dx + y) + 2x + 3y² dy/dx = 0
Solve for dy/dx :
x cos(xy) dy/dx + y cos(xy) + 2x + 3y² dy/dx = 0
(x cos(xy) + 3y²) dy/dx = - (y cos(xy) + 2x)
dy/dx = - (y cos(xy) + 2x) / (x cos(xy) + 3y²)
