To find the derivative of $5x$, we use the power rule for differentiation, which states that $\frac{d}{dx}(cx^n) = cnx^{n-1}$. Step 1: Identify the constant and the variable term. In $5x$, the constant is $5$ and the variable term is $x$, which can be written as $x^1$. Step 2: Apply the power rule. $$ \frac{d}{dx}(5x^1) = 5 \cdot 1 \cdot x^{1-1} $$ $$ \frac{d}{dx}(5x) = 5 \cdot x^0 $$ Step 3: Simplify the expression. Since $x^0 = 1$ for any non-zero $x$: $$ \frac{d}{dx}(5x) = 5 \cdot 1 $$ $$ \frac{d}{dx}(5x) = 5 $$ The derivative of $5x$ is $\boxed{5}$.