Final answer:
To prove the equation dx/dy = 2 - x/y, we first simplify the equation x/xy = ln(2/xy) by dividing x by xy and rewriting the right side using the property of logarithms. We then take the derivative of both sides with respect to y and equate the derivatives to solve for dx/dy. The final result is dx/dy = 2 - x/y.
Step-by-step explanation:
Given the equation x/xy = ln(2/xy), we need to prove that dx/dy = 2 - x/y.
To begin, we can simplify the left side of the equation. Dividing x by xy means we can cancel out the x, leaving us with 1/y. On the right side, we can rewrite ln(2/xy) as ln(2) - ln(xy) using the property that the logarithm of a division is the difference of the logarithms.
So, our equation becomes 1/y = ln(2) - ln(xy).
To proceed, we can take the derivative of both sides with respect to y. The derivative of 1/y is -1/y^2, and the derivative of ln(2) - ln(xy) is 0 - (1/xy) * (dx/dy), which simplifies to -1/xy * (dx/dy).
Now we can equate the two derivatives: -1/y^2 = -1/xy * (dx/dy).
Cross-multiplying gives us -xy * (-1/y^2) = dx/dy.
Simplifying further, we get dx/dy = 2 - x/y, as desired.