Calculate the gradient of the curve (x^{3} + y^{3} – 2xy = 11) at (2, -1).
Explanation
(x^{3} + y^{3} – 2xy = 11)
Differentiating implicitly,
(3x^{2} + 3y^{2} frac{mathrm d y}{mathrm d x} – 2y – 2x frac{mathrm d y}{mathrm d x} = 0)
((3y^{2} – 2x) frac{mathrm d y}{mathrm d x} = 2y – 3x^{2})
(frac{mathrm d y}{mathrm d x} = frac{2y – 3x^{2}}{3y^{2} – 2x})
At (2, -1) , Gradient = (frac{2(-1) – 3(2^{2})}{3(-1)^{2} – 2(2)})
= (frac{-2 – 12}{3 – 4})
= (frac{-14}{-1} = 14)