https://file.aiquickdraw.com/mathbot/file/1726598867188ht1sgqzi.jpg

The function provided is:

\[
f(x) = -2x^2 + 13x
\]

We are tasked with finding the derivative \( f'(x) \), and then calculating specific values of this derivative at \( x = 4 \), \( x = 17 \), and \( x = -3 \).

### Step 1: Find the derivative of \( f(x) \)

The derivative of \( f(x) \) with respect to \( x \) can be calculated using basic differentiation rules. For \( f(x) = -2x^2 + 13x \):

\[
f'(x) = \frac{d}{dx} (-2x^2 + 13x)
\]

Using the power rule:
- The derivative of \( -2x^2 \) is \( -4x \),
- The derivative of \( 13x \) is \( 13 \).

So, the derivative is:

\[
f'(x) = -4x + 13
\]

### Step 2: Compute the specific values

1. \( f'(4) \):
   \[
   f'(4) = -4(4) + 13 = -16 + 13 = -3
   \]

2. \( f'(17) \):
   \[
   f'(17) = -4(17) + 13 = -68 + 13 = -55
   \]

3. \( f'(-3) \):
   \[
   f'(-3) = -4(-3) + 13 = 12 + 13 = 25
   \]

### Conclusion:

- \( f'(4) = -3 \)
- \( f'(17) = -55 \)
- \( f'(-3) = 25 \)

Let me know if you need further clarifications! Here are five questions to deepen your understanding:

1. How do you apply the power rule when differentiating functions?
2. Why does the derivative of a quadratic function result in a linear expression?
3. What is the significance of calculating the derivative at specific points?
4. How does the slope of the function \( f(x) = -2x^2 + 13x \) change as \( x \) increases?
5. Can you explain why the derivative of a constant is zero?

**Tip:** Always check your differentiation by revisiting basic rules like the power rule and chain rule to ensure your results are accurate.

Use technology to find f'(4), f'(17), f'(-3) for the given function f(x) = -2x^2 + 13x where the derivative exists.

Learn how to compute the derivative of the function f(x) = -2x^2 + 13x using the power rule. Find specific values of f'(4), f'(17), and f'(-3) with a detailed solution. Suitable for high school students studying calculus.

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