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.