We are given the equation:

$-2x^2 + 64 = -3y + 4x - 3y^2$

Step 1: Rearrange the equation

First, we simplify and rearrange the given equation. Combine like terms on the right-hand side:

$-2x^2 + 64 = 4x - 3y - 3y^2$

Step 2: Implicit Differentiation

To find the derivative at a particular point, we need to perform implicit differentiation on both sides of the equation with respect to $x$, keeping in mind that $y$ is also a function of $x$ (i.e., $y = y(x)$).

Starting with the left-hand side:

$\frac{d}{dx}(-2x^2 + 64) = -4x$

Now for the right-hand side. We differentiate each term:

$\frac{d}{dx}(4x) = 4$

For $-3y$, we apply the chain rule:

$\frac{d}{dx}(-3y) = -3 \frac{dy}{dx}$

For $-3y^2$, we also apply the chain rule:

$\frac{d}{dx}(-3y^2) = -6y \frac{dy}{dx}$

Step 3: Set up the derivative equation

Now combine everything:

$-4x = 4 - 3 \frac{dy}{dx} - 6y \frac{dy}{dx}$

Step 4: Solve for $\frac{dy}{dx}$

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

$-4x - 4 = -3 \frac{dy}{dx} - 6y \frac{dy}{dx}$

Factor out $\frac{dy}{dx}$ on the right-hand side:

$-4x - 4 = \frac{dy}{dx}(-3 - 6y)$

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

$\frac{dy}{dx} = \frac{-4x - 4}{-3 - 6y}$

Step 5: Evaluate at the point $(5, 1)$

Now substitute $x = 5$ and $y = 1$ into the derivative:

$\frac{dy}{dx} = \frac{-4(5) - 4}{-3 - 6(1)}$

Simplify the numerator and denominator:

$\frac{dy}{dx} = \frac{-20 - 4}{-3 - 6} = \frac{-24}{-9} = \frac{24}{9} = \frac{8}{3}$

Final Answer:

The derivative $\frac{dy}{dx}$ at the point $(5, 1)$ is:

$\boxed{\frac{8}{3}}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does implicit differentiation work when both $x$ and $y$ are variables?
2. Can you explain the chain rule in more detail?
3. What are the typical applications of implicit differentiation?
4. How would you find the second derivative from this result?
5. How does implicit differentiation differ from explicit differentiation?

Tip: Always keep track of both variables when differentiating implicitly—one mistake can lead to errors in the solution!