The derivative of the function $f(x) = \frac{1}{x}$ can be found using standard differentiation rules. This function can be rewritten as:

$f(x) = x^{-1}$

Now, apply the power rule for derivatives, which states:

$\frac{d}{dx}(x^n) = n \cdot x^{n-1}$

For $f(x) = x^{-1}$, the exponent $n = -1$, so:

$f'(x) = -1 \cdot x^{-2} = -\frac{1}{x^2}$

Thus, the derivative of $f(x) = \frac{1}{x}$ is:

$f'(x) = -\frac{1}{x^2}$

Would you like a deeper explanation of this process or have any further questions?

Here are some related questions:

1. What is the derivative of $\frac{1}{x^2}$?
2. How do you find the derivative of a product involving $\frac{1}{x}$?
3. What is the second derivative of $\frac{1}{x}$?
4. How does the derivative of $\frac{1}{x}$ behave as $x$ approaches zero?
5. Can you explain the chain rule for composite functions involving $\frac{1}{x}$?

Tip: When applying the power rule, always reduce the exponent by 1 and multiply by the original exponent!