To find $y'$ (the derivative of $y$ with respect to $x$), we need to differentiate each term of the given function:

$y = -3e^{4x^2} - 5x + 3$

Step-by-step differentiation:

1. First term: $-3e^{4x^2}$

   We apply the chain rule. The derivative of $e^{u}$ is $e^u \cdot \frac{du}{dx}$, where $u = 4x^2$.

   $\frac{d}{dx}\left(-3e^{4x^2}\right) = -3 \cdot e^{4x^2} \cdot \frac{d}{dx}(4x^2)$ The derivative of $4x^2$ is $8x$, so this becomes: $-3e^{4x^2} \cdot 8x = -24x e^{4x^2}$

2. Second term: $-5x$

   The derivative of $-5x$ is simply: $-5$

3. Third term: $3$

   The derivative of a constant is 0: $0$

Putting it all together:

$y' = -24x e^{4x^2} - 5$

So, the derivative of $y$ with respect to $x$ is:

$y' = -24x e^{4x^2} - 5$

Would you like to dive deeper into any specific part of the solution? Here are some related questions:

1. What is the chain rule and how does it apply in this example?
2. How would the result change if the exponent in the first term was different (e.g., $e^{2x^2}$)?
3. What would happen if we had an additional term involving $x^3$ in the function?
4. How do you handle derivatives of higher powers of $x$ in exponential functions?
5. What is the general formula for differentiating an exponential function like $e^{ax^2}$?

Tip: When applying the chain rule, always remember to differentiate the inner function first, then multiply by the derivative of the outer function!