To find the derivative of the function

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

we differentiate term by term using basic differentiation rules, including the product rule for the term $x e^x$.

Step 1: Differentiate each term

• The derivative of the constant $1$ is $0$.
• The derivative of $2x$ is $2$.
• The term $x e^x$ requires the product rule because it is the product of two functions of $x$: $x$ and $e^x$.

The product rule states that if $y = u(x)v(x)$, then:

$\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

Here, $u(x) = x$ and $v(x) = e^x$, so:

[ \frac{d}{dx} [x e^x] = \frac{d}{dx} [x] \cdot e^x + x \cdot \frac{d}{dx} [e^x] ]

This becomes:

$\frac{d}{dx} [x e^x] = 1 \cdot e^x + x \cdot e^x = e^x + x e^x$

Step 2: Combine the results

Now, differentiate the entire function:

$\frac{dy}{dx} = \frac{d}{dx} \left( 1 + 2x + x e^x \right)$ $\frac{dy}{dx} = 0 + 2 + (e^x + x e^x)$ $\frac{dy}{dx} = 2 + e^x + x e^x$

Conclusion:

The derivative of the function $y = 1 + 2x + x e^x$ is:

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

Would you like more details or further clarifications?

Here are 5 related questions:

1. How would the derivative change if the function had $e^{2x}$ instead of $e^x$?
2. What is the second derivative of $y = 1 + 2x + x e^x$?
3. How can you apply the product rule to more complex products?
4. How do you find the derivative of the function $y = x^2 e^x$?
5. What is the derivative of $e^x$ divided by $x$?

Tip: When differentiating products of functions, always apply the product rule to avoid missing terms.