Final answer:
To calculate Dx/dy for the equation Y^x = X^y, take the natural logarithm of both sides, differentiate implicitly, and solve for Dx/dy, resulting in Dx/dy = ln(X) / ln(Y), assuming Y and X are constants.
Step-by-step explanation:
To calculate Dx/dy given that Y^x = X^y, we first acknowledge that this is an equation involving two variables, x and y. Solving such an equation analytically can be challenging due to its non-standard form. However, we can apply a technique involving logarithms and implicit differentiation to find the derivative of x with respect to y.
First, we would take the natural logarithm of both sides of the equation to obtain:
ln(Y^x) = ln(X^y)
This simplifies to:
x ln(Y) = y ln(X)
Now, we differentiate both sides with respect to y, keeping in mind that x is a function of y (and thus using the chain rule):
d/dy [x ln(Y)] = d/dy [y ln(X)]
We get:
(dx/dy) ln(Y) + (x/Y) (dY/dy) = ln(X) + (y/X) (dX/dy)
Assuming X and Y are constants, their derivatives with respect to y would be zero, simplifying the equation to:
(dx/dy) ln(Y) = ln(X)
Finally, we solve for dx/dy by dividing both sides by ln(Y):
dx/dy = ln(X) / ln(Y)
This gives us the derivative of x with respect to y, assuming Y and X are constants and Y does not equal 1 (as ln(1) = 0, which would make the denominator zero).