# How do you determine #(d^2y)/(dx^2)# given #x-y=(x+y)^2#?

By the chain rule we get

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To find (\frac{{d^2y}}{{dx^2}}) given (x - y = (x + y)^2), you first need to solve for (y) in terms of (x). Then, differentiate the resulting expression for (y) twice with respect to (x).

Starting with (x - y = (x + y)^2), you solve for (y) to get:

[y = x - (x + y)^2]

[y = x - (x^2 + 2xy + y^2)]

[y = x - x^2 - 2xy - y^2]

[y + y^2 = x - x^2 - 2xy]

[y^2 + y + x^2 + 2xy - x = 0]

Now, differentiate (y) with respect to (x) to find (y'). Then, differentiate (y') with respect to (x) to find (y'').

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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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