To find $f'(-2)$, we first need to determine the derivative of $F(x)$.

Step 1: Differentiate $F(x) = 3x^2 - x$

Using basic differentiation rules: $F'(x) = \frac{d}{dx}(3x^2) - \frac{d}{dx}(x)$ $F'(x) = 6x - 1$

Step 2: Evaluate $F'(x)$ at $x = -2$

$F'(-2) = 6(-2) - 1$ $F'(-2) = -12 - 1 = -13$

Final Answer:

$F'(-2) = -13$

Would you like a deeper explanation or have further questions? Here are related questions you might find interesting:

1. What is the significance of $F'(x)$ in a function?
2. How do you apply the power rule for derivatives?
3. What does $F'(x)$ represent geometrically on a graph?
4. How can we find higher-order derivatives of $F(x)$?
5. What are some applications of derivatives in physics or economics?

Tip: Always verify your derivative using multiple methods (e.g., first principles) for complex functions.