Final answer:
To find dy/dx by implicit differentiation for the equation x² = (5x + 4y) / (5x - 4y), apply the derivative to both sides of the equation, use the quotient rule for the right side, and solve for dy/dx.
Step-by-step explanation:
To find dy/dx by implicit differentiation for the equation x² = (5x + 4y) / (5x - 4y), we'll apply the derivative to both sides concerning x, remembering to use the quotient rule when differentiating the right side of the equation. Here's a step by step approach:
- Differentiate both sides with respect to x: 2x = d/dx [(5x + 4y) / (5x - 4y)].
- Apply the quotient rule on the right side, which is d/dx [u/v] = (v(du/dx) - u(dv/dx)) / v².
- Identify u = 5x + 4y and v = 5x - 4y and find du/dx and dv/dx accordingly.
- Solve for dy/dx.
The calculating steps involve algebra and derivatives, which lead to an expression for dy/dx as a function of x and y.