To solve for $\frac{dy}{dx}$, given that:

$y = 4u - 4 \quad \text{and} \quad u = -x^2 + 3,$

we will follow these steps:

Step 1: Differentiate $y$ with respect to $u$

From the equation $y = 4u - 4$, differentiate $y$ with respect to $u$: $\frac{dy}{du} = 4.$

Step 2: Differentiate $u$ with respect to $x$

From the equation $u = -x^2 + 3$, differentiate $u$ with respect to $x$: $\frac{du}{dx} = -2x.$

Step 3: Apply the chain rule

Using the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, we get: $\frac{dy}{dx} = 4 \cdot (-2x) = -8x.$

Thus, the final result is:

$\frac{dy}{dx} = -8x.$

Let me know if you would like more details or have any questions!

Here are some related questions:

1. How do you apply the chain rule in differentiation?
2. What is implicit differentiation, and how does it differ from the chain rule?
3. How do we differentiate polynomial functions?
4. What is the significance of $\frac{du}{dx}$ in composite functions?
5. Can the same approach be used to differentiate trigonometric functions?

Tip: Always identify the variables in your function and decide the appropriate rule of differentiation (e.g., chain rule, product rule) based on the relationships between them.