use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\\( 6 x ^ { 2 } y + 7 x y ^ { 2 } =…

use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\\( 6 x ^ { 2 } y + 7 x y ^ { 2 } = - 6 \\)\n\\( \\frac { d y } { d x } = \\)

use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\\( 6 x ^ { 2 } y + 7 x y ^ { 2 } = - 6 \\)\n\\( \\frac { d y } { d x } = \\)

Answer

Explanation:

Step1: Differentiate both sides

Differentiate (6x^{2}y + 7xy^{2}=-6) with respect to (x) using the product rule ((uv)^\prime = u^\prime v+uv^\prime). For (6x^{2}y), let (u = 6x^{2}) and (v=y). Then ((6x^{2}y)^\prime=12xy + 6x^{2}\frac{dy}{dx}). For (7xy^{2}), let (u = 7x) and (v = y^{2}). Then ((7xy^{2})^\prime=7y^{2}+14xy\frac{dy}{dx}). The derivative of (-6) is (0). So (12xy + 6x^{2}\frac{dy}{dx}+7y^{2}+14xy\frac{dy}{dx}=0).

Step2: Solve for (\frac{dy}{dx})

Group the terms with (\frac{dy}{dx}): ((6x^{2}+14xy)\frac{dy}{dx}=-12xy - 7y^{2}). Factor out (2x) from (6x^{2}+14xy) (get (2x(3x + 7y))) and factor out (-y) from (-12xy-7y^{2}) (get (-y(12x + 7y))). Then (\frac{dy}{dx}=\frac{-12xy - 7y^{2}}{6x^{2}+14xy}=\frac{-y(12x + 7y)}{2x(3x + 7y)}).

Answer:

(\frac{-y(12x + 7y)}{2x(3x + 7y)})