We are tasked with finding the slope of the tangent line to the curve given by:

$-3x^2 + 4xy - 4y^3 = -35$

at the point $(3, 2)$, under a modified differentiation rule on Z Planet. The differentiation rule on Z Planet states:

$\frac{d}{dx}x^n = n^2x^{n-1}$

This differs from the usual Earth rule where $\frac{d}{dx}x^n = nx^{n-1}$.

Step 1: Differentiate implicitly

We need to apply implicit differentiation to the given equation $-3x^2 + 4xy - 4y^3 = -35$, using the Z Planet rule for differentiating terms involving $x$.

Differentiating each term with respect to $x$:

1. First term: $-3x^2$

   • Using the Z Planet rule for differentiating $x^2$, we get: $\frac{d}{dx}(-3x^2) = -3 \cdot 2^2 x^{2-1} = -12x$

2. Second term: $4xy$

   • For the product $4xy$, we use the product rule: $\frac{d}{dx}(4xy) = 4 \left( \frac{d}{dx}(x) \cdot y + x \cdot \frac{dy}{dx} \right)$ Here, $\frac{d}{dx}(x) = 1$, so this becomes: $4(y + x \frac{dy}{dx})$

3. Third term: $-4y^3$

   • Since this term only involves $y$, we differentiate it using the chain rule: $\frac{d}{dx}(-4y^3) = -4 \cdot 3y^2 \cdot \frac{dy}{dx} = -12y^2 \cdot \frac{dy}{dx}$

4. Constant term: $-35$

   • The derivative of a constant is 0: $\frac{d}{dx}(-35) = 0$

Step 2: Combine the results

Now, we put everything together. The differentiated equation becomes:

$-12x + 4(y + x \frac{dy}{dx}) - 12y^2 \frac{dy}{dx} = 0$

Simplifying this:

$-12x + 4y + 4x \frac{dy}{dx} - 12y^2 \frac{dy}{dx} = 0$

Step 3: Solve for $\frac{dy}{dx}$

Rearrange the equation to solve for $\frac{dy}{dx}$:

$4x \frac{dy}{dx} - 12y^2 \frac{dy}{dx} = 12x - 4y$

Factor out $\frac{dy}{dx}$:

$\frac{dy}{dx} (4x - 12y^2) = 12x - 4y$

Solve for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{12x - 4y}{4x - 12y^2}$

Step 4: Evaluate at the point $(3, 2)$

Substitute $x = 3$ and $y = 2$ into the expression for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{12(3) - 4(2)}{4(3) - 12(2^2)} = \frac{36 - 8}{12 - 12(4)} = \frac{28}{12 - 48} = \frac{28}{-36} = -\frac{7}{9}$

Final Answer:

The slope of the tangent line to the curve at the point $(3, 2)$ is:

$\boxed{-\frac{7}{9}}$

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Would you like further details or have any questions on this?

Relative Questions:

1. How do we handle product rules under different differentiation rules?
2. Why is implicit differentiation useful in cases like this?
3. What are the implications of modifying basic differentiation rules like $\frac{d}{dx} x^n$?
4. How would the solution change if we were given a different modified differentiation rule?
5. What techniques are useful for solving implicit differentiation problems more generally?

Tip:

When applying modified differentiation rules, always focus on ensuring consistency in applying the rule throughout, especially in product and chain rule cases.