To differentiate the function $y = x^2 e^{2x+3}$, we'll use the product rule and the chain rule.

Step 1: Recognize the structure

The function is a product of two functions:

• $u(x) = x^2$
• $v(x) = e^{2x+3}$

So, the product rule tells us: $\frac{d}{dx}(uv) = u'v + uv'$

Step 2: Differentiate each function

1. Differentiate $u(x) = x^2$: $u'(x) = 2x$

2. Differentiate $v(x) = e^{2x+3}$:

   • First, we need to differentiate the exponent $2x+3$, which gives $2$.
   • So, by the chain rule: $v'(x) = e^{2x+3} \cdot 2 = 2e^{2x+3}$

Step 3: Apply the product rule

Now we can apply the product rule: $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$ Substitute the derivatives we found: $\frac{dy}{dx} = (2x)(e^{2x+3}) + (x^2)(2e^{2x+3})$

Step 4: Simplify the expression

Factor out $e^{2x+3}$: $\frac{dy}{dx} = e^{2x+3} \left( 2x + 2x^2 \right)$

So, the derivative is: $\frac{dy}{dx} = e^{2x+3} (2x + 2x^2)$

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Would you like more details on this process or have any questions?

Here are 5 related questions:

1. How does the chain rule work when differentiating an exponential function?
2. What are other methods for differentiating products of functions besides the product rule?
3. How do you simplify expressions like $e^{2x+3} (2x + 2x^2)$?
4. What are some applications of the product rule in real-world problems?
5. How would you differentiate a function involving trigonometric and exponential terms?

Tip: Always identify the basic structure of the function (product, quotient, chain rule) before proceeding with differentiation!