https://file.aiquickdraw.com/mathbot/file/1727033317647b7tzpkko.jpg

To solve the problem, we are given the function:

\[
f(x) = 2 + 4x - 5x^2
\]

and asked to find \( f'(-4) \), which is the value of the derivative of the function at \( x = -4 \).

### Steps:
1. **Differentiate the function \( f(x) \) with respect to \( x \):**

   \[
   f'(x) = \frac{d}{dx} \left( 2 + 4x - 5x^2 \right)
   \]

   - The derivative of a constant (2) is 0.
   - The derivative of \( 4x \) is 4.
   - The derivative of \( -5x^2 \) is \( -10x \) (using the power rule).

   Thus, the derivative is:

   \[
   f'(x) = 4 - 10x
   \]

2. **Substitute \( x = -4 \) into the derivative:**

   \[
   f'(-4) = 4 - 10(-4)
   \]
   \[
   f'(-4) = 4 + 40 = 44
   \]

### Final Answer:
\[
f'(-4) = 44
\]

Would you like any further details on this problem?

---

Here are 5 related questions you might want to consider:
1. How do you apply the power rule for differentiation?
2. What are the steps for differentiating polynomial functions?
3. How do you find the second derivative of a function?
4. How do derivatives relate to the slope of the tangent line at a point?
5. Can you interpret the physical meaning of a derivative in real-world problems?

**Tip:** Always check your derivative calculations by simplifying each term carefully before evaluating at a specific point.

Learn how to differentiate the function f(x) = 2 + 4x - 5x^2 and find the derivative at x = -4. This step-by-step guide covers key calculus concepts such as the power rule for differentiation. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation