You are here:

Master Calculus! Get instant help on “I Aced Calculus AP” App. Hundreds of flashcards and practice questions at your fingertips. Download now on the App Store and Google Play.

Derivative of $\dfrac{x}{3}$

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.

NEED QUICK
CALC HELP?
Download the I Aced Calculus App today!
ALL Calc Topics, 1000+ of PRACTICE questions