# How do you differentiate #-y=lnxy-xy#?

this is a bit easier [administratively] if you use the Implicit Function Theorem...

there are some pretty simple ways to intuit the Implicit Function Theorem

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To differentiate the equation -y = ln(xy) - xy with respect to both x and y, you can use implicit differentiation.

Differentiate both sides of the equation with respect to x: d/dx(-y) = d/dx(ln(xy) - xy)

This yields: -dy/dx = (1/xy)(y + x(dy/dx)) - (y + x(dy/dx))

Now, differentiate both sides of the equation with respect to y: d/dy(-y) = d/dy(ln(xy) - xy)

This results in: -1 = (1/x)(y + x(dy/dx)) + x

Now, solve for dy/dx by isolating it in the equation obtained from differentiating with respect to x. This gives: dy/dx = -xy/(x + 1 + ln(xy))

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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