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Let Yx=Xy. Calculate Dx / dy. Explain Your Answer Fully.

User Skd
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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).

User Luoluo
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