1. **State the problem:** Find the derivative $\frac{dy}{dx}$ implicitly from the equation $6x - 5y^2 - 3xy - x = 11$ and evaluate it at the point $(1,3)$. 2. **Rewrite the equation:** $$6x - 5y^2 - 3xy - x = 11$$ Simplify the left side: $$6x - x - 5y^2 - 3xy = 11$$ $$5x - 5y^2 - 3xy = 11$$ 3. **Differentiate both sides with respect to $x$ implicitly:** Use the product rule for $-3xy$ and chain rule for $y^2$: $$\frac{d}{dx}(5x) - \frac{d}{dx}(5y^2) - \frac{d}{dx}(3xy) = \frac{d}{dx}(11)$$ 4. **Calculate each derivative:** $$5 - 5 \cdot 2y \frac{dy}{dx} - \left(3y + 3x \frac{dy}{dx}\right) = 0$$ 5. **Simplify:** $$5 - 10y \frac{dy}{dx} - 3y - 3x \frac{dy}{dx} = 0$$ 6. **Group terms with $\frac{dy}{dx}$ and constants:** $$5 - 3y = 10y \frac{dy}{dx} + 3x \frac{dy}{dx}$$ 7. **Factor out $\frac{dy}{dx}$:** $$5 - 3y = \left(10y + 3x\right) \frac{dy}{dx}$$ 8. **Solve for $\frac{dy}{dx}$:** $$\frac{dy}{dx} = \frac{5 - 3y}{10y + 3x}$$ 9. **Evaluate at the point $(1,3)$:** $$\frac{dy}{dx} = \frac{5 - 3 \times 3}{10 \times 3 + 3 \times 1} = \frac{5 - 9}{30 + 3} = \frac{-4}{33}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = -\frac{4}{33}}$$