## Explanation

To find the derivative of the function $f(x) = \frac{1}{x}$, we will use basic differentiation rules. Here are the steps:

### Step 1: Recognize the Function Form

The function $\frac{1}{x}$ can be rewritten using a negative exponent:

$f(x) = x^{-1}$### Step 2: Apply the Power Rule

The power rule for differentiation states that if $f(x) = x^n$, then $f'(x) = n x^{n-1}$. For this function, $n = -1$:

$f'(x) = (-1) x^{-1 - 1}$### Step 3: Simplify the Expression

Now, simplify the expression:

$f'(x) = - x^{-2}$This can be written back in fractional form:

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

So, the **derivative** of $\frac{1}{x}$ is:

### Important Note

**Memorizing** this result can be very useful as it frequently appears in calculus problems. Basically:

Understanding this differentiation helps in solving complex calculus problems more efficiently.