Final answer:
To find dy/dx for the equation x² + y² = 4xy, one must use implicit differentiation and then isolate dy/dx to solve for it. The result is dy/dx = (4y - 2x) / (2y - 4x).
Step-by-step explanation:
The student asked to find dy/dx for the equation x² + y² = 4xy. To solve this, we need to use implicit differentiation because y is not expressed explicitly as a function of x. Differentiating both sides of the equation with respect to x gives us:
-
- 2x + 2y(dy/dx) = 4x(dy/dx) + 4y
-
- Now, we rearrange the terms to isolate dy/dx on one side:
-
- 2y(dy/dx) - 4x(dy/dx) = 4y - 2x
-
- (2y - 4x)(dy/dx) = 4y - 2x
-
- Divide both sides by (2y - 4x) to solve for dy/dx:
-
- dy/dx = (4y - 2x) / (2y - 4x)
This gives us the derivative of y with respect to x for the given equation.