The derivative expression you need to find is:

$\frac{d}{dx} \left( \frac{3 - x e^x}{x + e^x} \right)$

This is a quotient, so we will use the quotient rule for differentiation. The quotient rule states that for a function of the form $\frac{f(x)}{g(x)}$, the derivative is given by:

$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Step-by-step solution:

1. Identify $f(x)$ and $g(x)$:

   • $f(x) = 3 - x e^x$
   • $g(x) = x + e^x$

2. Find $f'(x)$:

   • The derivative of $3$ is $0$.
   • The derivative of $- x e^x$ uses the product rule: $d(x e^x)/dx = e^x + x e^x$.

   So: $f'(x) = - (e^x + x e^x) = -e^x - x e^x$

3. Find $g'(x)$:

   • The derivative of $x$ is $1$.
   • The derivative of $e^x$ is $e^x$.

   So: $g'(x) = 1 + e^x$

4. Apply the quotient rule: $\frac{d}{dx} \left( \frac{3 - x e^x}{x + e^x} \right) = \frac{( -e^x - x e^x)(x + e^x) - (3 - x e^x)(1 + e^x)}{(x + e^x)^2}$

5. Simplify the expression: Expand the numerator and combine like terms. The resulting expression would likely simplify further, but leaving it in expanded form gives you the exact steps from the quotient rule application.

Would you like me to simplify the expression further or keep it in this form? Also, feel free to ask any follow-up questions or request more details!

Additional questions to explore:

1. How do you apply the quotient rule in general?
2. What is the product rule and how is it applied in this context?
3. What are the derivative rules for exponential functions?
4. How can we simplify the numerator further?
5. How does the chain rule apply when differentiating products involving exponential functions?

Tip:

When applying the quotient rule, ensure that you carefully expand the terms in both the numerator and denominator before simplifying to avoid mistakes.