https://file.aiquickdraw.com/mathbot/file/1724337845544rk9ey98l.jpg

The function provided in the image is:

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

To find the derivative of this function, \( f'(x) \), and then evaluate it at \( x = -1 \), we follow these steps:

### Step 1: Differentiate \( f(x) \)
1. Differentiate \(-(x + 1)\):
   \[
   \frac{d}{dx}[-(x + 1)] = -1
   \]
2. Differentiate \(\frac{3}{x - 2}\) using the chain rule:
   \[
   \frac{d}{dx}\left(\frac{3}{x - 2}\right) = 3 \cdot \frac{d}{dx}\left(\frac{1}{x - 2}\right)
   \]
   The derivative of \(\frac{1}{x - 2}\) is:
   \[
   \frac{d}{dx}\left(\frac{1}{x - 2}\right) = -\frac{1}{(x - 2)^2}
   \]
   Therefore,
   \[
   \frac{d}{dx}\left(\frac{3}{x - 2}\right) = -\frac{3}{(x - 2)^2}
   \]

So, the derivative \( f'(x) \) is:
\[
f'(x) = -1 - \frac{3}{(x - 2)^2}
\]

### Step 2: Evaluate \( f'(x) \) at \( x = -1 \)
\[
f'(-1) = -1 - \frac{3}{(-1 - 2)^2} = -1 - \frac{3}{(-3)^2} = -1 - \frac{3}{9} = -1 - \frac{1}{3} = -\frac{4}{3}
\]

Thus, the derivative of the function at \( x = -1 \) is \( f'(-1) = -\frac{4}{3} \).

Would you like more details on this process, or do you have any other questions?

### Related Questions:
1. How do you apply the chain rule in differentiation?
2. What is the significance of the derivative of a function?
3. How do you find the critical points of a function?
4. What are the steps to differentiate rational functions?
5. How do you interpret the derivative in real-world applications?

### Tip:
Always remember to simplify the function as much as possible before differentiating. This can often make the differentiation process easier.

Learn how to find the derivative of the function -(x + 1) + 3/(x - 2) and evaluate it at x = -1. This problem involves differentiation using the chain rule and is suitable for college-level students. Explore step-by-step solutions and related mathematical concepts.

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