To find the equation of the tangent line to the curve at $x = 1$, we need to follow these steps:

1. Implicit Differentiation: Differentiate the equation $3y^3 + xy - y = 81x^4$ with respect to $x$ to find $\frac{dy}{dx}$.

   Given: $3y^3 + xy - y = 81x^4$

   Differentiating both sides with respect to $x$: $\frac{d}{dx}(3y^3) + \frac{d}{dx}(xy) - \frac{d}{dx}(y) = \frac{d}{dx}(81x^4)$

   Using the chain rule and product rule: $3 \cdot 3y^2 \cdot \frac{dy}{dx} + y + x \frac{dy}{dx} - \frac{dy}{dx} = 324x^3$

   Simplify the terms to group $\frac{dy}{dx}$: $(9y^2 + x - 1)\frac{dy}{dx} = 324x^3 - y$

   Solving for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{324x^3 - y}{9y^2 + x - 1}$

2. Evaluate $y$ when $x = 1$: Substitute $x = 1$ into the original equation to find the corresponding $y$-value.

   $3y^3 + (1)(y) - y = 81(1)^4$ $3y^3 = 81$ $y^3 = 27$ $y = 3$

   So, when $x = 1$, $y = 3$.

3. Calculate the Slope of the Tangent Line: Substitute $x = 1$ and $y = 3$ into the derivative $\frac{dy}{dx}$ to find the slope at that point.

   $\frac{dy}{dx} = \frac{324(1)^3 - 3}{9(3)^2 + 1 - 1}$ $= \frac{324 - 3}{81} = \frac{321}{81} = 4$

   So, the slope of the tangent line at $(1, 3)$ is 4.

4. Equation of the Tangent Line: Use the point-slope form of the line equation, $y - y_1 = m(x - x_1)$, with $(x_1, y_1) = (1, 3)$ and $m = 4$.

   $y - 3 = 4(x - 1)$ $y = 4x - 4 + 3$ $y = 4x - 1$

So, the equation of the tangent line is: $y = 4x - 1$

Would you like further details on any step, or do you have additional questions?

Here are some related questions to explore:

1. How would you find the normal line to the curve at $x = 1$?
2. What if $x$ and $y$ terms were higher powers? How would that affect differentiation?
3. How can you verify the point $(1,3)$ lies on the original curve?
4. How does implicit differentiation differ from explicit differentiation in this problem?
5. Could we use a different method to find $\frac{dy}{dx}$ here?

Tip: When using implicit differentiation, always remember to apply the product rule where necessary, especially when both $x$ and $y$ are in the same term.