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The image shows a problem asking to determine the derivative of a given function \( w(s) \) using the **Quotient Rule**.

The function provided is:

\[
w(s) = \frac{6 + 5e^s}{3e^s + 4}
\]

### Quotient Rule:

If you have a function of the form:

\[
w(s) = \frac{f(s)}{g(s)}
\]

Then the derivative is:

\[
w'(s) = \frac{f'(s)g(s) - f(s)g'(s)}{(g(s))^2}
\]

### Step-by-step Solution:

1. **Identify \( f(s) \) and \( g(s) \):**
   - \( f(s) = 6 + 5e^s \)
   - \( g(s) = 3e^s + 4 \)

2. **Differentiate \( f(s) \) and \( g(s) \):**
   - \( f'(s) = \frac{d}{ds} \left(6 + 5e^s \right) = 5e^s \)
   - \( g'(s) = \frac{d}{ds} \left(3e^s + 4 \right) = 3e^s \)

3. **Apply the Quotient Rule:**

\[
w'(s) = \frac{(5e^s)(3e^s + 4) - (6 + 5e^s)(3e^s)}{(3e^s + 4)^2}
\]

4. **Simplify the expression:**
   - First, expand the terms in the numerator:
     \[
     (5e^s)(3e^s + 4) = 15e^{2s} + 20e^s
     \]
     \[
     (6 + 5e^s)(3e^s) = 18e^s + 15e^{2s}
     \]
   - Now subtract the terms:
     \[
     15e^{2s} + 20e^s - (18e^s + 15e^{2s}) = 15e^{2s} - 15e^{2s} + 20e^s - 18e^s = 2e^s
     \]

5. **Final simplified derivative:**

\[
w'(s) = \frac{2e^s}{(3e^s + 4)^2}
\]

This is the derivative of the function \( w(s) \).

---

Would you like further details or explanations?

### Related Questions:
1. How does the Quotient Rule differ from the Product Rule?
2. Can we apply the Chain Rule to simplify certain steps in the derivative?
3. How do exponentials behave when differentiated?
4. What are the applications of the Quotient Rule in physics or engineering?
5. How would the derivative change if there was a different exponent in \( e^s \)?

### Tip:
Always simplify expressions step by step to avoid errors when applying rules like the Quotient Rule.

Use the Quotient Rule to determine the derivative of the function w(s) = (6 + 5e^s) / (3e^s + 4)

Learn how to calculate the derivative of the function w(s) = (6 + 5e^s) / (3e^s + 4) using the Quotient Rule. Step-by-step solution provided with simplified final answer. Suitable for Calculus 1 or 2 students.

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