\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (-x) = -1} \end{align*}\]

Let’s learn how to find the derivative of the function \( f(x) = -x \). Don’t worry—we’ll keep it simple and easy to understand!

### What is a Derivative?

A **derivative** tells us how a function changes as the input \( x \) changes. Think of it like measuring how fast something is moving. In math, it shows us the rate at which \( y \) is changing with respect to \( x \).

### Step-by-Step Explanation

#### Step 1: Write the Function

\[\begin{align*} f(x) = -x \end{align*}\]

#### Step 2: Identify the Power of \( x \)

The variable \( x \) has an exponent of 1 (since \( x = x^1 \)):

\[\begin{align*} f(x) = -1 \times x^1 \end{align*}\]

#### Step 3: Recall the Power Rule

The **Power Rule **helps us find the derivative of \( x \) raised to a power:

\[\begin{align*} \frac{d}{dx} \left( x^n \right) = n \times x^{n-1} \end{align*}\]

where \( n \) is any real number.

#### Step 4: Apply the Power Rule

Apply the Power Rule to \( -1 \times x^1 \):

\[\begin{align*} \frac{d}{dx} (-1 \times x^1) = -1 \times \frac{d}{dx} (x^1) \end{align*}\]

Using the Power Rule on \( x^1 \):

\[\begin{align*} \frac{d}{dx} (x^1) = 1 \times x^{1-1} = 1 \times x^0 \end{align*}\]

Since \( x^0 = 1 \):

\[\begin{align*} 1 \times x^0 = 1 \times 1 = 1 \end{align*}\]

Now, multiply back by \(-1\):

\[\begin{align*} -1 \times 1 = -1 \end{align*}\]

#### Step 5: Write the Derivative

So, the derivative of \( f(x) = -x \) is:

\[\begin{align*} \frac{d}{dx} (-x) = -1 \end{align*}\]

### Conclusion

The rate at which \( -x \) changes with respect to \( x \) is constant and equal to \(-1\). This means that the slope of the line \( y = -x \) is always \(-1\).

\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (-x) = -1} \end{align*}\]