# How do you find the derivative of # e^(xy)=x/y#?

Start from the equation:

As the first member is always positive, so is the second, and we can take the logarithm of both sides:

using the properties of logarithms this becomes:

Keep in mind that:

so we have:

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To find the derivative of the equation e^(xy) = x/y, we can use implicit differentiation.

First, we differentiate both sides of the equation with respect to x:

d/dx(e^(xy)) = d/dx(x/y)

Using the chain rule, the derivative of e^(xy) with respect to x is:

d/dx(e^(xy)) = e^(xy) * (y + xy')

Next, we differentiate x/y with respect to x using the quotient rule:

d/dx(x/y) = (y * 1 - x * 1/y^2) / y^2

Simplifying this expression, we get:

d/dx(x/y) = (y - x/y^2) / y^2

Now, equating the derivatives obtained from both sides of the equation, we have:

e^(xy) * (y + xy') = (y - x/y^2) / y^2

To find y', we can isolate it by rearranging the equation:

y' = [(y - x/y^2) / y^2] / e^(xy) - xy

Therefore, the derivative of e^(xy) = x/y with respect to x is:

y' = [(y - x/y^2) / y^2] / e^(xy) - xy

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