2. Let $f$ be the function defined by $f(x)=-\frac{1}{x+10}$. Which of the following statements is/are true? I. $f$ is differentiable at $x=-10$. II. $f$ is continuous at $x=-10$. III. $f$ has a vertical asymptote at $x=-10$. III only I only I, II, \& III II only I \& III

Step 1 ：We are given the function \(f(x)=-\frac{1}{x+10}\).

Step 2 ：We need to check which of the following statements is/are true: I. \(f\) is differentiable at \(x=-10\). II. \(f\) is continuous at \(x=-10\). III. \(f\) has a vertical asymptote at \(x=-10\).

Step 3 ：To check if the function is differentiable at \(x=-10\), we need to check if the derivative of the function exists at \(x=-10\). The derivative of a function at a point exists if the function is smooth (has no breaks, holes, or sharp corners) at that point.

Step 4 ：To check if the function is continuous at \(x=-10\), we need to check if the limit of the function as \(x\) approaches \(-10\) exists and is equal to the value of the function at \(x=-10\).

Step 5 ：To check if the function has a vertical asymptote at \(x=-10\), we need to check if the limit of the function as \(x\) approaches \(-10\) from the left or the right is infinity or negative infinity.

Step 6 ：The derivative of the function \(f(x)=-\frac{1}{x+10}\) is \((x + 10)^{-2}\), which is undefined at \(x=-10\). Therefore, the function is not differentiable at \(x=-10\).

Step 7 ：The limit of the function as \(x\) approaches \(-10\) from the left is infinity and from the right is negative infinity. Since these two limits are not equal, the function is not continuous at \(x=-10\).

Step 8 ：However, since the limit of the function as \(x\) approaches \(-10\) from the left or the right is infinity or negative infinity, the function has a vertical asymptote at \(x=-10\).

Step 9 ：Final Answer: The correct statement is III only. So, the final answer is \(\boxed{\text{III only}}\).