To prove the equation x/xy = ln(2/xy) and show that dx/dy = 2 - x/y, we can simplify the equation and solve for dx/dy. By dividing both sides, applying the chain rule, and canceling out common terms, we arrive at the result dx/dy = 2 - x/y.
To prove this statement, we need to simplify the given equation and solve for dx/dy.
Given: x/xy = ln(2/xy)
Divide both sides by x: 1/y = ln(2/xy)
Take the derivative of both sides with respect to y: d(1/y)/dy = d(ln(2/xy))/dy
Apply the chain rule on the left side: -1/y^2(dy/dy) = (1/(2/xy))(d(2/xy)/dy)
Simplify the right side: -1/y^2(dy/dy) = (1/(2/xy))(d(2/xy)/dy) = 1/(2yx)(d(2/xy)/dy) = 1/(2yx)(-2/x^2)(dx/dy) = -1/(x^2y^2)(dx/dy)
Cancel out -1 on both sides: y^2(dx/dy) = x^2y^2(dy/dy)
Divide both sides by y^2: dx/dy = x^2(dy/dy)
Cancel out y^2 on the right side: dx/dy = x^2
Simplify further: dx/dy = 2 - x/y
Therefore, we have proved that dx/dy = 2 - x/y using the given equation and mathematical rules.