To find dy/dx using implicit differentiation for the equation x² + 8xy + y² = 8, follow these steps:
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Start with the given equation:
x² + 8xy + y² = 8
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Differentiate both sides with respect to x. Remember, when you differentiate terms involving y, you will need to apply the chain rule and include dy/dx:
The derivative of x² is 2x.
The derivative of 8xy uses the product rule:
If u = 8x and v = y, then d(uv)/dx = u(dv/dx) + v(du/dx), resulting in:
8y + 8x(dy/dx)
The derivative of y² is 2y(dy/dx).
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Now put it all together:
2x + (8y + 8x(dy/dx)) + 2y(dy/dx) = 0
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Rearranging the equation, isolate all terms that contain dy/dx:
8x(dy/dx) + 2y(dy/dx) = -2x – 8y
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Factor out dy/dx:
dy/dx (8x + 2y) = -2x – 8y
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Now, solve for dy/dx:
dy/dx = \frac{-2x – 8y}{8x + 2y}
So, the derivative of y with respect to x, dy/dx, is given by:
dy/dx = \frac{-2x – 8y}{8x + 2y}
This result tells you how y changes with respect to x at any point on the curve defined by the given equation.