Equation: x^2+y^2 =2x +4y. Given that dy/dt =6, find dx/dt when (x,y) = (1+√2,2+√3).

To find dx/dt, we need to differentiate both sides of the equation x^2 + y^2 = 2x + 4y implicitly with respect to t (time), using the chain rule.

Step 1: Differentiate both sides of the equation with respect to t:
d/dt (x^2 + y^2) = d/dt (2x + 4y)

Step 2: Use the chain rule on each term:
2x * dx/dt + 2y * dy/dt = 2 * dx/dt + 4 * dy/dt

Step 3: Substitute the given values to the equation:
2(1+√2) * dx/dt + 2(2+√3) * 6 = 2 * dx/dt + 4 * 6

Step 4: Simplify the equation:
2√2 * dx/dt + 4√3 = 2 * dx/dt + 24

Step 5: Move all terms containing dx/dt to one side of the equation:
2√2 * dx/dt - 2 * dx/dt = 24 - 4√3

Step 6: Combine like terms:
(2√2 - 2) * dx/dt = 24 - 4√3

Step 7: Divide both sides by (2√2 - 2) to isolate dx/dt:
dx/dt = (24 - 4√3) / (2√2 - 2)

So, dx/dt is equal to (24 - 4√3) / (2√2 - 2) when (x, y) = (1+√2, 2+√3).