# What is the derivative of # (x^3)-(xy)+(y^3)=1#?

Differentiating:

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To find the derivative of the implicit function (x^3 - xy + y^3 = 1), differentiate both sides of the equation with respect to (x), applying the chain rule where necessary.

[ \frac{{d}}{{dx}}(x^3 - xy + y^3) = \frac{{d}}{{dx}}(1) ]

[ 3x^2 - (x\frac{{dy}}{{dx}} + y) - (xy\frac{{dy}}{{dx}} + 3y^2\frac{{dy}}{{dx}}) = 0 ]

[ 3x^2 - x\frac{{dy}}{{dx}} - y - xy\frac{{dy}}{{dx}} - 3y^2\frac{{dy}}{{dx}} = 0 ]

[ 3x^2 - y - (x + 3y^2)\frac{{dy}}{{dx}} = 0 ]

[ (x + 3y^2)\frac{{dy}}{{dx}} = 3x^2 - y ]

[ \frac{{dy}}{{dx}} = \frac{{3x^2 - y}}{{x + 3y^2}} ]

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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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