To find the derivative of \( y = \left( e^{4x} - 2 \right)^3 \), we will apply the **chain rule**.

### Steps:

1. **Outer function**: 
   The outer function is \( u^3 \), where \( u = e^{4x} - 2 \). The derivative of \( u^3 \) with respect to \( u \) is \( 3u^2 \).

2. **Inner function**: 
   The inner function is \( u = e^{4x} - 2 \). The derivative of \( e^{4x} \) with respect to \( x \) is \( 4e^{4x} \), and the derivative of the constant \( -2 \) is 0.

### Derivative Calculation:

\[
\frac{dy}{dx} = 3\left( e^{4x} - 2 \right)^2 \cdot \frac{d}{dx}\left( e^{4x} - 2 \right)
\]
\[
\frac{dy}{dx} = 3\left( e^{4x} - 2 \right)^2 \cdot 4e^{4x}
\]
\[
\frac{dy}{dx} = 12e^{4x}\left( e^{4x} - 2 \right)^2
\]

So, the derivative is:
\[
\frac{dy}{dx} = 12e^{4x}\left( e^{4x} - 2 \right)^2
\]

Would you like a detailed explanation of any step, or do you have further questions?

Here are some related questions for practice:
1. What is the derivative of \( y = \left( e^{3x} - 1 \right)^4 \)?
2. How would you differentiate \( y = \left( e^{2x} + 3 \right)^5 \)?
3. What is the second derivative of \( y = \left( e^{4x} - 2 \right)^3 \)?
4. How does the chain rule apply to composite functions?
5. Can you generalize the derivative of \( y = \left( f(x) \right)^n \)?

**Tip**: When differentiating composite functions, always identify and differentiate the outer and inner functions separately, then multiply using the chain rule!

Learn how to find the derivative of y = (e^4x - 2)^3 using the chain rule. This calculus problem involves differentiating composite functions and applying the chain rule to exponential expressions. Suitable for students in Grades 11-12 or those studying Calculus 1.

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