# How do you find #dy/dx# by implicit differentiation of #x^3-xy+y^2=4#?

For the second term we apply the product rule;

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To find ( \frac{dy}{dx} ) by implicit differentiation of ( x^3 - xy + y^2 = 4 ), you differentiate each term with respect to ( x ) and then solve for ( \frac{dy}{dx} ).

Differentiating each term with respect to ( x ), we get: [ \frac{d}{dx}(x^3) - \frac{d}{dx}(xy) + \frac{d}{dx}(y^2) = \frac{d}{dx}(4) ]

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

Rearranging terms and solving for ( \frac{dy}{dx} ), we get: [ \frac{dy}{dx} = \frac{x - 3x^2 + y}{2y - x} ]

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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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