# How do you differentiate #y=(x-y)^2/(x+y)#?

Given:

Multiply both sides by

Expand the square:

Combine like terms:

Divide both sides by

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To differentiate the function ( y = \frac{{(x - y)^2}}{{x + y}} ) with respect to ( x ), we'll use the quotient rule and implicit differentiation.

Let ( u = (x - y)^2 ) and ( v = x + y ). Then,

( \frac{{dy}}{{dx}} = \frac{{v \frac{{du}}{{dx}} - u \frac{{dv}}{{dx}}}}{{v^2}} )

First, differentiate ( u ) and ( v ) with respect to ( x ):

( \frac{{du}}{{dx}} = 2(x - y)(1 - \frac{{dy}}{{dx}}) - 1 )

( \frac{{dv}}{{dx}} = 1 + \frac{{dy}}{{dx}} )

Now, substitute these into the quotient rule formula:

( \frac{{dy}}{{dx}} = \frac{{(x + y)(2(x - y)(1 - \frac{{dy}}{{dx}}) - 1) - (x - y)^2(1 + \frac{{dy}}{{dx}})}}{{(x + y)^2}} )

Now, solve for ( \frac{{dy}}{{dx}} ).

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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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