To differentiate the given function $y = (4x - 2)^2 (3 - x^5)^2$, we can apply both the product rule and the chain rule.

Step-by-Step Solution:

1. Product Rule: The function is a product of two terms, $(4x - 2)^2$ and $(3 - x^5)^2$. So, we apply the product rule: $\frac{dy}{dx} = \frac{d}{dx} \left( (4x - 2)^2 \right) \cdot (3 - x^5)^2 + (4x - 2)^2 \cdot \frac{d}{dx} \left( (3 - x^5)^2 \right)$

2. Differentiate the first term $(4x - 2)^2$ using the chain rule:

   • Let $u = 4x - 2$, then: $\frac{d}{dx} \left( u^2 \right) = 2u \cdot \frac{du}{dx} = 2(4x - 2) \cdot 4 = 8(4x - 2)$

3. Differentiate the second term $(3 - x^5)^2$ using the chain rule:

   • Let $v = 3 - x^5$, then: $\frac{d}{dx} \left( v^2 \right) = 2v \cdot \frac{dv}{dx} = 2(3 - x^5) \cdot (-5x^4) = -10x^4(3 - x^5)$

4. Combine using the product rule:

   • The derivative is: $\frac{dy}{dx} = 8(4x - 2)(3 - x^5)^2 + (4x - 2)^2(-10x^4)(3 - x^5)$

Final Answer:

$\frac{dy}{dx} = 8(4x - 2)(3 - x^5)^2 - 10x^4(4x - 2)^2(3 - x^5)$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you apply the product rule to more complex functions?
2. What is the chain rule, and why is it important in this context?
3. How do higher-order derivatives of this function behave?
4. Can this function's derivative be simplified further?
5. How would you approach integration for this kind of expression?

Tip: When applying the product rule, be mindful of simplifying each term after differentiation to avoid mistakes later on.