The given function is:

$y = (x + 1)^2 e^x$

This is a combination of a polynomial $(x + 1)^2$ and the exponential function $e^x$. Let's break it down and compute a few important properties of the function.

1. First Derivative (dy/dx):

To differentiate $y$ with respect to $x$, we'll use the product rule since it's a product of two functions: $f(x) = (x + 1)^2$ and $g(x) = e^x$. The product rule states:

$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• $f(x) = (x + 1)^2$
• $g(x) = e^x$

Now, compute the derivatives of each:

• $f'(x) = 2(x + 1)$
• $g'(x) = e^x$

Now apply the product rule:

$\frac{dy}{dx} = 2(x + 1) e^x + (x + 1)^2 e^x$

Factor out $e^x$:

$\frac{dy}{dx} = e^x \left[ 2(x + 1) + (x + 1)^2 \right]$

Simplify the expression inside the brackets:

$\frac{dy}{dx} = e^x \left[ 2(x + 1) + (x + 1)(x + 1) \right]$ $= e^x \left[ 2(x + 1) + (x^2 + 2x + 1) \right]$ $= e^x \left[ x^2 + 4x + 3 \right]$

Thus, the first derivative is:

$\frac{dy}{dx} = e^x (x^2 + 4x + 3)$

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Would you like to explore the second derivative or other details about this function?

Here are some additional related questions you may find useful:

1. How do we find the critical points of this function?
2. What is the second derivative $\frac{d^2y}{dx^2}$?
3. Can you graph this function?
4. How do you find the inflection points of this function?
5. What is the behavior of the function as $x \to \infty$?

Tip: When differentiating a product of functions, always use the product rule and simplify step by step to avoid mistakes.