Use implicit differentiation to determine $$ \frac{d y}{d x} $$ for the equation $$ 3 x y^{2}=x-10 $$

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1. Start with the given equation:
   $$
   3xy^2 = x - 10
   $$

2. Differentiate both sides with respect to $$x$$. For the left-hand side, apply the product rule:
   $$
   \frac{d}{dx}(3xy^2) = 3\frac{d}{dx}(xy^2) = 3\left(\frac{d}{dx}(x)y^2 + x\frac{d}{dx}(y^2)\right)
   $$
   Notice that:
   $$
   \frac{d}{dx}(x) = 1 \quad \text{and} \quad \frac{d}{dx}(y^2) = 2y\frac{dy}{dx}
   $$
   So the derivative of the left-hand side becomes:
   $$
   3\left(y^2 + 2xy\frac{dy}{dx}\right) = 3y^2 + 6xy\frac{dy}{dx}
   $$

3. Differentiate the right-hand side:
   $$
   \frac{d}{dx}(x-10) = 1
   $$

4. Set the derivatives equal:
   $$
   3y^2 + 6xy\frac{dy}{dx} = 1
   $$

5. Solve for $$\frac{dy}{dx}$$:
   - Subtract $$3y^2$$ from both sides:
     $$
     6xy\frac{dy}{dx} = 1 - 3y^2
     $$
   - Divide both sides by $$6xy$$:
     $$
     \frac{dy}{dx} = \frac{1 - 3y^2}{6xy}
     $$

The derivative is:
$$
\frac{dy}{dx} = \frac{1 - 3y^2}{6xy}
$$
$$ \frac{dy}{dx} = \frac{1 - 3y^2}{6xy} $$

Use implicit differentiation to determine \( \frac{d y}{d x} \) for the equation \( 3 x y^{2}=x-10 \)

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