For the curve given by 4x^2 +y^2 = 48 + 2xy show that dy/dx =y-4x/y-x
4x^2 +y^2 = 48 + 2xy
8x + 2yy' = 2y + 2xy'
y'(2y-2x) = 2y-8x
...
To find the derivative dy/dx for the curve given by the equation 4x^2 + y^2 = 48 + 2xy, we can use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x, treating y as a function of x:
d/dx (4x^2 + y^2) = d/dx (48 + 2xy)
Step 2: Apply the chain rule on the left-hand side. The derivative of y^2 with respect to x can be found as 2y * dy/dx.
8x + 2y * dy/dx = 0 + 2x * dy/dx + 2y
Step 3: Simplify the equation by combining like terms:
8x + 2y * dy/dx = 2x * dy/dx + 2y
Step 4: Move all terms with dy/dx to one side of the equation:
2y * dy/dx - 2x * dy/dx = -8x + 2y
Step 5: Factor out dy/dx from the left-hand side:
dy/dx * (2y - 2x) = -8x + 2y
Step 6: Divide both sides of the equation by (2y - 2x):
dy/dx = (-8x + 2y) / (2y - 2x)
Step 7: Simplify the expression on the right-hand side:
dy/dx = (-8x + 2y) / 2(y - x)
Step 8: Divide both the numerator and denominator by 2:
dy/dx = (-4x + y) / (y - x)
Therefore, the derivative dy/dx for the given curve is (y - 4x) / (y - x).