The derivative of $\arctan\left(\frac{1}{x}\right)$ can be found using the chain rule. Here's how we proceed step by step:

1. Let $y = \arctan\left(\frac{1}{x}\right)$.

2. The derivative of $\arctan(u)$ with respect to $u$ is $\frac{1}{1+u^2}$, so: $\frac{d}{dx} \arctan\left( \frac{1}{x} \right) = \frac{1}{1 + \left(\frac{1}{x}\right)^2} \cdot \frac{d}{dx}\left(\frac{1}{x}\right)$

3. Simplify the expression inside the arctangent's derivative: $\frac{1}{1 + \left(\frac{1}{x}\right)^2} = \frac{1}{1 + \frac{1}{x^2}} = \frac{x^2}{x^2 + 1}$

4. Now, find the derivative of $\frac{1}{x}$, which is $-\frac{1}{x^2}$.

5. Multiply the results: $\frac{d}{dx} \arctan\left( \frac{1}{x} \right) = \frac{x^2}{x^2 + 1} \cdot \left(-\frac{1}{x^2}\right)$

6. Simplify: $\frac{d}{dx} \arctan\left( \frac{1}{x} \right) = -\frac{1}{x^2 + 1}$

Thus, the derivative of $\arctan\left(\frac{1}{x}\right)$ is: $\boxed{-\frac{1}{x^2 + 1}}$

Would you like further details on the steps? Here are some related questions:

1. What is the derivative of $\arctan(x)$?
2. How does the chain rule work with composite functions?
3. Can you apply the chain rule to more complex trigonometric functions?
4. What is the second derivative of $\arctan\left(\frac{1}{x}\right)$?
5. How can you apply implicit differentiation to inverse functions?

Tip: When differentiating functions with nested parts (like $\arctan$ with an inner function), always break it down using the chain rule carefully!