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The function given is:

\[
f(x) = \frac{3 - xe^x}{x + e^x}
\]

To differentiate this, we will apply the **quotient rule**, which is used when differentiating a ratio of two functions. The quotient rule states:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) u'(x) - u(x) v'(x)}{(v(x))^2}
\]

Here, \( u(x) = 3 - xe^x \) and \( v(x) = x + e^x \).

### Step-by-step differentiation:

1. **Differentiate \( u(x) = 3 - xe^x \):**
   \[
   u'(x) = -e^x - xe^x
   \]
   (This uses the product rule for differentiating \( xe^x \), and the derivative of 3 is 0.)

2. **Differentiate \( v(x) = x + e^x \):**
   \[
   v'(x) = 1 + e^x
   \]

3. **Apply the quotient rule:**
   \[
   f'(x) = \frac{(x + e^x)(-e^x - xe^x) - (3 - xe^x)(1 + e^x)}{(x + e^x)^2}
   \]

4. **Simplify the numerator:**

   Distribute both terms in the numerator:

   \[
   (x + e^x)(-e^x - xe^x) = -xe^{2x} - x e^x - e^{2x} - e^x
   \]
   \[
   (3 - xe^x)(1 + e^x) = 3 + 3e^x - xe^x - xe^{2x}
   \]

   Now subtract the two expressions:

   \[
   \left(-xe^{2x} - x e^x - e^{2x} - e^x\right) - \left(3 + 3e^x - xe^x - xe^{2x}\right)
   \]

   Simplify further:

   \[
   -xe^{2x} - e^{2x} - x e^x - e^x - 3 - 3e^x + xe^x + xe^{2x}
   \]

   Combine like terms:

   \[
   -e^{2x} - 4e^x - 3
   \]

5. **Final derivative expression:**
   \[
   f'(x) = \frac{-e^{2x} - 4e^x - 3}{(x + e^x)^2}
   \]

This is the derivative of the given function.

Would you like more details or further explanations?

Here are 5 related questions you could explore:
1. How would you differentiate the same function using logarithmic differentiation?
2. Can you solve for critical points of \( f(x) \) using the derivative?
3. What is the significance of the quotient rule in calculus?
4. How does the presence of \( e^x \) influence the behavior of the function as \( x \) approaches infinity?
5. How can we verify the result of this derivative using an alternative method?

**Tip:** When simplifying expressions, be careful to track negative signs and distribution to avoid common algebraic mistakes.

Differentiate the function: f(x) = (3 - xe^x) / (x + e^x)

Learn how to differentiate the function f(x) = (3 - xe^x) / (x + e^x) using the quotient rule. This solution covers step-by-step differentiation, including the application of the quotient rule, and simplifies the derivative expression. Ideal for students in Grades 11-12 studying calculus.

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