https://file.aiquickdraw.com/mathbot/file/17259410855615vql3pg5.jpg

The provided image contains a function \( f(x) \) defined as:

\[
f(x) = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}
\]

We are tasked with finding \( f'(-2) \), which involves first finding the derivative \( f'(x) \) and then substituting \( x = -2 \).

### Step 1: Differentiate \( f(x) \)
We'll use the power rule for differentiation. First, rewrite the function in terms of negative exponents:

\[
f(x) = x^{-1} + x^{-2} + x^{-3}
\]

Now, differentiate each term:

\[
f'(x) = \frac{d}{dx}(x^{-1}) + \frac{d}{dx}(x^{-2}) + \frac{d}{dx}(x^{-3})
\]

Applying the power rule \( \frac{d}{dx}(x^n) = n x^{n-1} \), we get:

\[
f'(x) = -x^{-2} - 2x^{-3} - 3x^{-4}
\]

Which simplifies to:

\[
f'(x) = -\frac{1}{x^2} - \frac{2}{x^3} - \frac{3}{x^4}
\]

### Step 2: Substitute \( x = -2 \) into \( f'(x) \)

Now substitute \( x = -2 \) into the derivative:

\[
f'(-2) = -\frac{1}{(-2)^2} - \frac{2}{(-2)^3} - \frac{3}{(-2)^4}
\]

Simplifying each term:

- \( (-2)^2 = 4 \), so \( -\frac{1}{4} \)
- \( (-2)^3 = -8 \), so \( -\frac{2}{-8} = \frac{1}{4} \)
- \( (-2)^4 = 16 \), so \( -\frac{3}{16} \)

Now sum the terms:

\[
f'(-2) = -\frac{1}{4} + \frac{1}{4} - \frac{3}{16}
\]

Simplifying further:

\[
f'(-2) = 0 - \frac{3}{16} = -\frac{3}{16}
\]

### Final Answer:
\[
f'(-2) = -\frac{3}{16}
\]

Would you like further details or have any questions about this?

Here are some relative questions:
1. What is the general method for applying the power rule in derivatives?
2. How do you handle negative exponents when differentiating?
3. Can you explain the step-by-step process of simplifying expressions with fractions?
4. What is the importance of recognizing patterns in differentiation?
5. How can chain rule be used when functions are more complex?

**Tip:** Always rewrite functions with negative exponents to make differentiation easier with the power rule.

If f(x) = 1/x + 1/x^2 + 1/x^3, then what is f'(-2)?

This solution demonstrates how to differentiate the function f(x) = 1/x + 1/x^2 + 1/x^3 using the power rule and evaluates f'(-2) step-by-step. Learn how to handle negative exponents and apply calculus principles.

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