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!