https://file.aiquickdraw.com/mathbot/file/1726885324090rhrvj4xr.jpg

The question on the screen asks for the values of derivatives of the function \( f(x) = \frac{-2}{x} \), specifically for \( f'(2) \), \( f'(16) \), and \( f'(-4) \).

### Step 1: Find the derivative of the function \( f(x) = \frac{-2}{x} \)
We can differentiate \( f(x) \) using the power rule. Rewrite the function as:
\[
f(x) = -2x^{-1}
\]
Now, differentiate using the power rule:
\[
f'(x) = -2 \cdot (-1) x^{-2} = \frac{2}{x^2}
\]

### Step 2: Calculate \( f'(2) \)
Substitute \( x = 2 \) into \( f'(x) = \frac{2}{x^2} \):
\[
f'(2) = \frac{2}{2^2} = \frac{2}{4} = 0.5
\]

### Step 3: Calculate \( f'(16) \)
Substitute \( x = 16 \) into \( f'(x) \):
\[
f'(16) = \frac{2}{16^2} = \frac{2}{256} = 0.0078125
\]

### Step 4: Calculate \( f'(-4) \)
Substitute \( x = -4 \) into \( f'(x) \):
\[
f'(-4) = \frac{2}{(-4)^2} = \frac{2}{16} = 0.125
\]

### Answers:
- \( f'(2) = 0.5 \)
- \( f'(16) = 0.0078 \) (rounded to four decimal places)
- \( f'(-4) = 0.125 \)

Would you like a further breakdown of any step or more details?

---

Here are five related questions:
1. How do you find the derivative of a function using the power rule?
2. What happens when you take the derivative of a constant multiplied by a function?
3. Can the derivative of a function be negative, and how would that affect the graph?
4. How do you interpret the second derivative of a function?
5. Why does the derivative become smaller for larger values of \( x \) in the function \( f(x) = \frac{-2}{x} \)?

**Tip**: Always simplify the function into a form that is easier to differentiate (e.g., rewrite division as a power).

Use technology to find f'(2), f'(16), and f'(-4) for the function f(x) = -2/x, when the derivative exists.

Learn how to find the derivatives of the function f(x) = -2/x using the power rule. We calculate f'(2), f'(16), and f'(-4), showing detailed steps for each point. This calculus problem covers key concepts such as the power rule and handling negative exponents.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation