To find the derivative \( y' \) of the function \( y = (4x^4 - 8x^2 + 8)^4 \), we will use the chain rule. Let:

\[
u = 4x^4 - 8x^2 + 8
\]
so that
\[
y = u^4
\]

Now, applying the chain rule:

1. Differentiate \( y = u^4 \) with respect to \( u \):
   \[
   \frac{dy}{du} = 4u^3
   \]

2. Differentiate \( u = 4x^4 - 8x^2 + 8 \) with respect to \( x \):
   \[
   \frac{du}{dx} = 16x^3 - 16x
   \]

Now, we can combine these using \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \):

\[
y' = \frac{dy}{dx} = 4u^3 \cdot (16x^3 - 16x)
\]

Substitute back \( u = 4x^4 - 8x^2 + 8 \):

\[
y' = 4(4x^4 - 8x^2 + 8)^3 \cdot (16x^3 - 16x)
\]

Simplifying:

\[
y' = 64(4x^4 - 8x^2 + 8)^3 (x^3 - x)
\]

This is the derivative \( y' \) of the function.

Would you like further details or have any questions?

### Related Questions
1. How would you differentiate a function of the form \( (ax^n + bx^m + c)^k \)?
2. What is the significance of the chain rule in calculus?
3. How would you apply the power rule in differentiation?
4. Can this derivative be simplified further for specific values of \( x \)?
5. How does this method change if the exponent in the original problem is negative?

### Tip
Always simplify the inner function first when applying the chain rule to reduce complexity in the differentiation process.

Learn how to differentiate the function y = (4x^4 − 8x^2 + 8)^4 using the chain rule. This detailed solution explains key calculus concepts, including the chain rule and power rule, and provides step-by-step guidance for applying these to polynomial functions. Suitable for students in Grades 11-12 or early college.

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