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To find which option determines the derivative of \( f(x) = -5x^2 + 3x \), we need to apply the definition of the derivative using the limit process:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

### Step-by-Step Process:

1. **Find \( f(x+h) \):**
   \[
   f(x+h) = -5(x+h)^2 + 3(x+h)
   \]
   Expanding:
   \[
   f(x+h) = -5(x^2 + 2xh + h^2) + 3(x+h)
   \]
   \[
   = -5x^2 - 10xh - 5h^2 + 3x + 3h
   \]

2. **Subtract \( f(x) \) from \( f(x+h) \):**
   \[
   f(x+h) - f(x) = (-5x^2 - 10xh - 5h^2 + 3x + 3h) - (-5x^2 + 3x)
   \]
   Simplifying:
   \[
   = -10xh - 5h^2 + 3h
   \]

3. **Divide by \( h \):**
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{-10xh - 5h^2 + 3h}{h}
   \]
   Simplifying further:
   \[
   = -10x - 5h + 3
   \]

4. **Take the limit as \( h \to 0 \):**
   \[
   \lim_{h \to 0} (-10x - 5h + 3) = -10x + 3
   \]

Thus, the derivative of \( f(x) \) is \( f'(x) = -10x + 3 \).

### Now, match this process to the options provided in the image:

The correct expression is the one that applies the definition of the derivative:
\[
\frac{-5(x+h)^2 + 3(x+h) - (-5x^2 + 3x)}{h}
\]
This matches **Option 2**.

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

### Related Questions:
1. What is the definition of a derivative and how is it applied?
2. How do you expand binomials like \( (x+h)^2 \)?
3. What is the geometric interpretation of the derivative?
4. How would the derivative change if the function were cubic instead of quadratic?
5. Why do we take the limit as \( h \to 0 \) in the derivative definition?

### Tip:
Always simplify the function \( f(x+h) - f(x) \) carefully before dividing by \( h \) and taking the limit.

If f(x) = -5x^2 + 3x, which of the following determines the derivative of f(x)?

Learn how to determine the derivative of the function f(x) = -5x^2 + 3x using the limit definition of the derivative. This page covers essential calculus concepts including derivatives and limits, suitable for introductory college-level calculus courses.

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