The problem asks to find the derivative of the function $y = \frac{1}{x}$ with respect to $x$.

1. Problem Description

The problem asks to find the derivative of the function $y = \frac{1}{x}$ with respect to $x$.

2. Solution Steps

We are given $y = \frac{1}{x}$.

First, rewrite the function using a negative exponent:

$y = x^{-1}$

Now, we apply the power rule for differentiation:

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

Applying the power rule to $y = x^{-1}$, we have:

$\frac{dy}{dx} = (-1)x^{-1-1}$

$\frac{dy}{dx} = -1x^{-2}$

$\frac{dy}{dx} = -x^{-2}$

$\frac{dy}{dx} = -\frac{1}{x^2}$

3. Final Answer

The derivative of $y = \frac{1}{x}$ with respect to $x$ is $\frac{dy}{dx} = -\frac{1}{x^2}$.