To find the derivative of $y = (x^3 + x^2 + 1)^4$, we can apply the chain rule. Here's the process:

Step-by-step differentiation:

1. Outer function: The outer function is $u^4$, where $u = x^3 + x^2 + 1$.

   • The derivative of $u^4$ with respect to $u$ is $4u^3$.

2. Inner function: The inner function is $u = x^3 + x^2 + 1$.

   • The derivative of $x^3 + x^2 + 1$ with respect to $x$ is $3x^2 + 2x$.

Applying the chain rule:

$y' = 4(x^3 + x^2 + 1)^3 \cdot (3x^2 + 2x)$

This is the derivative of $y$.

Would you like further details or have any questions?

Here are some related questions:

1. What is the chain rule and why is it used here?
2. How would the derivative change if the power were different, say $(x^3 + x^2 + 1)^5$?
3. Can you apply the chain rule to trigonometric functions similarly?
4. How do we differentiate more complex nested functions using the chain rule?
5. What happens to the derivative if we simplify the inner function before differentiating?

Tip: Always double-check your derivative using basic rules like the chain and product rules!