Given the function \( y = 2x e^x \), you are asked to find \( y^n \), which typically refers to the \( n \)-th derivative of \( y \) with respect to \( x \).

### Steps to Calculate the \( n \)-th Derivative

1. **First Derivative \( y' \) (or \( y^{(1)} \)):**
   \[
   y' = \frac{d}{dx}(2x e^x)
   \]
   We apply the product rule:
   \[
   y' = 2 \left( \frac{d}{dx}(x) \cdot e^x + x \cdot \frac{d}{dx}(e^x) \right) = 2(e^x + x e^x) = 2e^x(1 + x)
   \]

2. **Second Derivative \( y'' \) (or \( y^{(2)} \)):**
   \[
   y'' = \frac{d}{dx}\left( 2e^x(1 + x) \right)
   \]
   Again, using the product rule:
   \[
   y'' = 2 \left( \frac{d}{dx}(e^x) \cdot (1 + x) + e^x \cdot \frac{d}{dx}(1 + x) \right) = 2 \left( e^x(1 + x) + e^x \right) = 2e^x(2 + x)
   \]

3. **Third Derivative \( y''' \) (or \( y^{(3)} \)):**
   \[
   y''' = \frac{d}{dx}\left( 2e^x(2 + x) \right)
   \]
   Again, using the product rule:
   \[
   y''' = 2 \left( e^x(2 + x) + e^x \right) = 2e^x(3 + x)
   \]

### General Pattern for the \( n \)-th Derivative:
Observing the pattern from the first few derivatives, the general form of the \( n \)-th derivative appears to be:
\[
y^{(n)} = 2e^x(n + x)
\]
This pattern holds because with each differentiation, the coefficient in front of \( e^x \) increases by 1.

### Final Answer:
\[
y^{(n)} = 2e^x(n + x)
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What happens if the exponent of \( e^x \) in the original function is changed?
2. How would the derivative process differ if there was an additional trigonometric factor?
3. Can you find the general derivative of a function of the form \( f(x) = x^n e^x \)?
4. What role does the product rule play in finding derivatives?
5. How does differentiation affect the exponential function?

**Tip:** When finding higher-order derivatives, look for patterns in the coefficients and terms to simplify the process.

Learn how to calculate the nth derivative of y = 2x e^x with step-by-step solutions. This problem covers differentiation, exponential functions, and the product rule in calculus. Suitable for advanced college-level mathematics.

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