The derivative of $4x$ is $\boxed{4}$.

To find the derivative of $4x$, we may apply the power rule:

\[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\]

In this case, \(4x\) can be rewritten as \(4x^1\), where the coefficient \(4\) is constant and \(x\) has an exponent of \(n=1\).

Solution

Using the power rule:

\[\begin{align*} \dfrac{d}{dx} (4x) &= 4 \cdot \dfrac{d}{dx} (x^1) \\ &= 4 \cdot (1)(x^{1 – 1}) \\ &= 4 \cdot 1 \cdot x^0 \\ &= 4 \cdot 1 \cdot 1 \\ &= \boxed{4} \end{align*}\]

Explanation

The derivative of \(4x\) is \(4\), which makes sense because the slope of any linear function of the form \(mx\) is simply the coefficient \(m\). Here, the coefficient is \(4\), so the slope and derivative of the function are both \(4\).

This result tells us that the rate of change of the function \(4x\) is constant. No matter the value of \(x\), the slope remains the same, which is a characteristic of linear functions.