Differentiate both sides of
xy ³ - 2x ²y ² = 0
with respect to x :
d(xy ³ - 2x ²y ²)/dx = d(0)/dx
d(xy ³)/dx - 2 d(x ²y ²)/dx = 0
By the product rule,
d(x)/dx y ³ + x d(y ³)/dx - 2 (d(x ²)/dx y ² + x ² d(y ²)/dx) = 0
By the chain rule,
y ³ + 3xy ² dy/dx - 2 (2xy ² + 2x ²y dy/dx) = 0
y ³ + 3xy ² dy/dx - 4xy ² - 4x ²y dy/dx = 0
y ³ - 4xy ² + (3xy ² - 4x ²y) dy/dx = 0
(3xy ² - 4x ²y) dy/dx = 4xy ² - y ³
dy/dx = (4xy ² - y ³) / (3xy ² - 4x ²y)
dy/dx = (4xy - y ²) / (3xy - 4x ²)
At the point (1, 2), the gradient is
dy/dx (1, 2) = (4×1×2 - 2²) / (3×1×2 - 4×1²) = 4/2 = 2
