To solve $F'(x)$ where $F(x) = \int_{x^2}^0 3t \, dt$, we apply the Fundamental Theorem of Calculus and chain rule.

Step-by-step solution:

1. Flip the integral's limits: $F(x) = \int_{x^2}^0 3t \, dt = -\int_0^{x^2} 3t \, dt$

2. Integrate $3t$ with respect to $t$: The integral of $3t$ is: $\int 3t \, dt = \frac{3t^2}{2}$

3. Evaluate the definite integral: $\int_0^{x^2} 3t \, dt = \left[\frac{3t^2}{2}\right]_0^{x^2} = \frac{3(x^2)^2}{2} - \frac{3(0)^2}{2} = \frac{3x^4}{2}$

   So, $F(x) = -\frac{3x^4}{2}$

4. Differentiate $F(x)$: Now, differentiate $F(x)$ using the chain rule: $F'(x) = -\frac{d}{dx} \left( \frac{3x^4}{2} \right)$ $F'(x) = -\frac{3 \cdot 4x^3}{2}$ $F'(x) = -6x^3$

5. Answer: The derivative is: $F'(x) = -6x^3$

   None of the given answers directly match. There might be a typo in the options, or they intended for another part of the question. Let me know if you want further clarification!

Would you like a detailed graph or additional examples?

Here are 5 follow-up questions:

1. What is the Fundamental Theorem of Calculus?
2. How do you handle negative integral limits?
3. Can you apply chain rule in other similar integrals?
4. What does the derivative $F'(x)$ represent graphically?
5. What happens when the function is more complex, e.g., $\int_{x^2}^{2x}$?

Tip: Always ensure to simplify and handle negative limits of integration carefully!