Find the gradient of (xy^{2} + x^{2} y = 4xy) at the point (1, 3).
Explanation
(xy^{2} + x^{2} y = 4xy)
Differentiating with respect to x,
(2xy frac{mathrm d y}{mathrm d x} + y^{2} + x^{2} frac{mathrm d y}{mathrm d x} + 2xy = 4x frac{mathrm d y}{mathrm d x} + 4y)
((2xy + x^{2} – 4x) frac{mathrm d y}{mathrm d x} = 4y – y^{2} – 2xy)
(frac{mathrm d y}{mathrm d x} = frac{4y^{2} – y^{2} – 2xy}{2xy + x^{2} – 4x})
At (1, 3), Gradient = (frac{4(3) – (3)^{2} – 2(1)(3)}{2(1)(3) + 1^{2} – 4(1)})
= (frac{-3}{3})
= -1