Simplify the expression $\frac{2x}{x^3 \cdot 2x}$.

1. Problem Description

Simplify the expression $\frac{2x}{x^3 \cdot 2x}$.

2. Solution Steps

First, we multiply the terms in the denominator:

$$$$

x^3 \cdot 2x = 2x^4

$$$$

Now the expression becomes:

$$$$

\frac{2x}{2x^4}

$$$$

Next, we can simplify the fraction by dividing both the numerator and the denominator by $2x$.

$$$$

\frac{2x}{2x^4} = \frac{2x \div 2x}{2x^4 \div 2x} = \frac{1}{x^3}

$$$$

3. Final Answer

The simplified expression is $\frac{1}{x^3}$.