dy dx of x square e to the power of (3x+2)

To find the derivative \(\frac{dy}{dx}\) of the function 

\[
y = x^2 e^{3x + 2}
\]

we'll apply the **product rule** and the **chain rule**.

### **Step 1: Recall the product rule**
\[
\frac{d}{dx} [f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)
\]

Here, \( f(x) = x^2 \) and \( g(x) = e^{3x + 2} \).

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### **Step 2: Differentiate \( f(x) \) and \( g(x) \)**

- \( f'(x) = \frac{d}{dx} (x^2) = 2x \)
- \( g'(x) = \frac{d}{dx} \left( e^{3x + 2} \right) \)

Using the chain rule, the derivative of \( e^{3x + 2} \) is:

\[
g'(x) = e^{3x + 2} \cdot \frac{d}{dx} (3x + 2) = 3 e^{3x + 2}
\]

---

### **Step 3: Apply the product rule**
\[
\frac{dy}{dx} = f'(x) \cdot g(x) + f(x) \cdot g'(x)
\]

Substitute the values:

\[
\frac{dy}{dx} = 2x e^{3x + 2} + x^2 \cdot 3 e^{3x + 2}
\]

---

### **Step 4: Simplify**
Factor out \( e^{3x + 2} \):

\[
\frac{dy}{dx} = e^{3x + 2} \left( 2x + 3x^2 \right)
\]

Thus, the final answer is:

\[
\boxed{\frac{dy}{dx} = e^{3x + 2} (3x^2 + 2x)}
\]

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Would you like more clarification on any step? 

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### **Related Questions:**  
1. How do you apply the product rule in general?  
2. Can you explain the chain rule in greater detail?  
3. What are other applications of exponential functions in calculus?  
4. How do you compute higher-order derivatives of exponential functions?  
5. What happens if the exponent of \( e \) contains a trigonometric function?  

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**Tip:** Always check for common factors when simplifying derivatives for cleaner expressions.

Learn how to compute the derivative of the function x^2 e^(3x+2) step-by-step using the product and chain rule. This solution covers key differentiation techniques for exponential functions, suitable for students in Grades 11-12.

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