# How do you implicitly differentiate #2= e^(xy^2-x^3y)-y^2x^3+y #?

Divide:

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To implicitly differentiate the equation ( 2 = e^{xy^2-x^3y} - y^2x^3 + y ) with respect to ( x ), you will differentiate each term individually while applying the chain rule and product rule where necessary.

Differentiating the first term ( e^{xy^2-x^3y} ) with respect to ( x ) using the chain rule: [ \frac{d}{dx} (e^{xy^2-x^3y}) = \frac{d}{dx}(xy^2-x^3y) \cdot e^{xy^2-x^3y} ]

Differentiating ( xy^2-x^3y ) with respect to ( x ): [ \frac{d}{dx}(xy^2-x^3y) = y^2 - 3x^2y ]

Therefore, the first term becomes: [ (e^{xy^2-x^3y}) \cdot (y^2 - 3x^2y) ]

Differentiating the second term ( -y^2x^3 ) with respect to ( x ): [ \frac{d}{dx} (-y^2x^3) = -3y^2x^2 ]

Differentiating the third term ( +y ) with respect to ( x ): [ \frac{d}{dx}(y) = \frac{dy}{dx} ]

So, the derivative of the equation with respect to ( x ) is: [ 0 = (e^{xy^2-x^3y}) \cdot (y^2 - 3x^2y) - 3y^2x^2 + \frac{dy}{dx} ]

This is the implicit differentiation of the given equation with respect to ( 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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