use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\n\\( y ^ { 2 } + 5 x ^ { 3 } = 8 y…

use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\n\\( y ^ { 2 } + 5 x ^ { 3 } = 8 y - 2 x ^ { 2 } \\)\n\n\\( \\frac { d y } { d x } = \\)\n\nquestion help: video message instructor\n\nsubmit question jump to answer

use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\n\\( y ^ { 2 } + 5 x ^ { 3 } = 8 y - 2 x ^ { 2 } \\)\n\n\\( \\frac { d y } { d x } = \\)\n\nquestion help: video message instructor\n\nsubmit question jump to answer

Answer

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (y^{2}+5x^{3}) and (8y - 2x^{2}) term - by - term. For (y^{2}), use the chain rule: (\frac{d}{dx}(y^{2})=2y\frac{dy}{dx}). For (5x^{3}), (\frac{d}{dx}(5x^{3}) = 15x^{2}). For (8y), (\frac{d}{dx}(8y)=8\frac{dy}{dx}). For (-2x^{2}), (\frac{d}{dx}(-2x^{2})=-4x). So, (2y\frac{dy}{dx}+15x^{2}=8\frac{dy}{dx}-4x).

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

Move all terms with (\frac{dy}{dx}) to one side: (2y\frac{dy}{dx}-8\frac{dy}{dx}=-4x - 15x^{2}). Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(2y - 8)=-4x - 15x^{2}). Then (\frac{dy}{dx}=\frac{-4x - 15x^{2}}{2y - 8}=\frac{15x^{2}+4x}{8 - 2y}).

Answer:

(\frac{15x^{2}+4x}{8 - 2y})