The derivative of the function $(x+2)(x-5)^{-1}$ with respect to $x$ is:

$\frac{1}{x - 5} - \frac{x + 2}{(x - 5)^2}$

Would you like any further details or clarification on this result?

Here are 5 related questions for further exploration:

1. How do we apply the product rule in differentiating more complex functions?
2. What are the key steps in differentiating rational functions?
3. How does chain rule apply to functions like $(x - 5)^{-1}$?
4. Can you explain the difference between implicit and explicit differentiation?
5. How can we simplify the result obtained from this derivative?

Tip: When differentiating a product of two functions, it's often useful to apply the product rule and simplify carefully to avoid common mistakes.