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We are tasked with finding the derivative of the function:

\[
y = \left( \frac{-2x - 1}{-4x + 2} \right)^5
\]

### Step-by-step Solution:

To differentiate this function, we will use the **chain rule** combined with the **quotient rule**. The outer function is raised to the power 5, so we'll apply the chain rule to that, and the inner function is a rational expression, so we'll apply the quotient rule to it.

1. **Applying the chain rule:**

   The outer function is \( u^5 \), where \( u = \frac{-2x - 1}{-4x + 2} \). Using the chain rule:
   
   \[
   y' = 5\left( \frac{-2x - 1}{-4x + 2} \right)^4 \cdot u'
   \]
   
   Now, we need to find \( u' \), the derivative of the inner function.

2. **Applying the quotient rule:**

   The quotient rule is:
   
   \[
   u' = \frac{(\text{denominator}) \cdot (\text{derivative of numerator}) - (\text{numerator}) \cdot (\text{derivative of denominator})}{(\text{denominator})^2}
   \]
   
   For the function \( u = \frac{-2x - 1}{-4x + 2} \):

   - Numerator: \( -2x - 1 \)
   - Denominator: \( -4x + 2 \)
   
   Derivatives:
   - Derivative of the numerator \( (-2x - 1) \) is \( -2 \).
   - Derivative of the denominator \( (-4x + 2) \) is \( -4 \).
   
   Now apply the quotient rule:
   
   \[
   u' = \frac{(-4x + 2)(-2) - (-2x - 1)(-4)}{(-4x + 2)^2}
   \]
   
   Simplifying:
   
   \[
   u' = \frac{(8x - 4) - (8x + 4)}{(-4x + 2)^2}
   \]
   
   \[
   u' = \frac{8x - 4 - 8x - 4}{(-4x + 2)^2}
   \]
   
   \[
   u' = \frac{-8}{(-4x + 2)^2}
   \]

3. **Final derivative:**

   Substituting \( u' \) into the chain rule expression:

   \[
   y' = 5\left( \frac{-2x - 1}{-4x + 2} \right)^4 \cdot \frac{-8}{(-4x + 2)^2}
   \]
   
   Simplifying the result:

   \[
   y' = \frac{-40(-2x - 1)^4}{(-4x + 2)^6}
   \]

This is the derivative of the given function.

---

Would you like a detailed breakdown of any part of this solution? Here are 5 related questions to further your understanding:

1. How does the quotient rule work when differentiating rational functions?
2. Can you apply the chain rule to different types of nested functions?
3. How do the exponents in chain rule applications affect the final result?
4. What simplifications can be made when applying the quotient rule?
5. How does the power of a function affect its derivative?

**Tip:** Always simplify rational expressions before applying the quotient rule to make the differentiation easier.

Find the derivative of y = ((-2x - 1)/(-4x + 2))^5.

Learn how to differentiate the function y = ((-2x - 1)/(-4x + 2))^5 using the chain rule and quotient rule. This problem involves applying advanced calculus concepts, including differentiating rational functions and using power rules.

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