1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the implicit function defined by the equation $$6x^2 - 5y^2 - 3xy - x = 11$$ at the point $(-1,3)$.\n\n2. **Use implicit differentiation:** Differentiate both sides of the equation with respect to $x$. Remember that $y$ is a function of $x$, so use the chain rule for terms involving $y$.\n\n3. **Differentiate each term:**\n- $\frac{d}{dx}(6x^2) = 12x$\n- $\frac{d}{dx}(-5y^2) = -10y \frac{dy}{dx}$ (chain rule)\n- $\frac{d}{dx}(-3xy) = -3\left(y + x \frac{dy}{dx}\right)$ (product rule)\n- $\frac{d}{dx}(-x) = -1$\n- $\frac{d}{dx}(11) = 0$\n\n4. **Write the differentiated equation:**\n$$12x - 10y \frac{dy}{dx} - 3y - 3x \frac{dy}{dx} - 1 = 0$$\n\n5. **Group terms with $\frac{dy}{dx}$ on one side:**\n$$-10y \frac{dy}{dx} - 3x \frac{dy}{dx} = -12x + 3y + 1$$\n\n6. **Factor out $\frac{dy}{dx}$:**\n$$\frac{dy}{dx}(-10y - 3x) = -12x + 3y + 1$$\n\n7. **Solve for $\frac{dy}{dx}$:**\n$$\frac{dy}{dx} = \frac{-12x + 3y + 1}{-10y - 3x}$$\n\n8. **Substitute the point $(-1,3)$:**\n$$\frac{dy}{dx} = \frac{-12(-1) + 3(3) + 1}{-10(3) - 3(-1)} = \frac{12 + 9 + 1}{-30 + 3} = \frac{22}{-27} = -\frac{22}{27}$$\n\n**Final answer:** $$\boxed{\frac{dy}{dx} = -\frac{22}{27}}$$