# How do you implicitly differentiate #x^2 y^3 − xy = 10#?

You can do it like this

Then apply the product rule and the chain rule:

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To implicitly differentiate the equation (x^2y^3 - xy = 10), follow these steps:

- Differentiate each term of the equation with respect to (x).
- Apply the product rule and chain rule when necessary.
- Solve for (\frac{dy}{dx}).

Differentiating each term: [ \frac{d}{dx}[x^2y^3] - \frac{d}{dx}[xy] = \frac{d}{dx}[10] ]

Apply the product rule and chain rule: [ 2xy^3 + x^2(3y^2 \cdot \frac{dy}{dx}) - (y + x\frac{dy}{dx}) = 0 ]

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

Thus, (\frac{dy}{dx} = \frac{y - 2xy^3}{3x^2y^2 - 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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