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.