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