The derivative of $\dfrac{x}{3}$ is $\boxed{\dfrac{1}{3}}$.
To find the derivative of $\dfrac{x}{3}$, we may apply the power rule:
\[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\]
In this case, $\dfrac{x}{3}$ can be rewritten as $\dfrac{1}{3} \cdot x^1$, where the coefficient $\dfrac13$ is constant and \(x\) has an exponent of \(n=1\).
Solution
Using the power rule:
\[\begin{align*} \dfrac{d}{dx} \dfrac{x}{3} &= \dfrac{1}{3} \cdot \dfrac{d}{dx} (x^1) \\ &= \dfrac{1}{3} \cdot (1)(x^{1 – 1}) \\ &= \dfrac{1}{3} \cdot 1 \cdot x^0 \\ &= \dfrac{1}{3} \cdot 1 \cdot 1 \\ &= \boxed{\dfrac{1}{3}} \end{align*}\]
Explanation
The derivative of \(\dfrac{x}{3}\) is \(\dfrac{1}{3}\), which makes sense because the slope of any linear function of the form \(mx\) is simply the coefficient \(m\). Here, the coefficient is \(\dfrac{1}{3}\), so the slope and derivative of the function are both \(\dfrac{1}{3}\).
This result tells us that the rate of change of the function \(\dfrac{x}{3}\) is constant. No matter the value of \(x\), the slope remains the same, which is a characteristic of linear functions.