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The problem asks to find the derivative of \( y = \left(3x^2 + \tan^{-1}(x)\right)^4 \), without simplifying the expression after taking all needed derivatives. Let's break it down step by step.

### Step 1: Use the Chain Rule

To differentiate \( y = \left(3x^2 + \tan^{-1}(x)\right)^4 \), we apply the chain rule. The outer function is \( u^4 \), where \( u = 3x^2 + \tan^{-1}(x) \). The derivative of \( u^4 \) is:

\[
\frac{dy}{dx} = 4u^3 \cdot \frac{du}{dx}
\]

### Step 2: Differentiate the Inner Function

Now we need to differentiate \( u = 3x^2 + \tan^{-1}(x) \).

- The derivative of \( 3x^2 \) is \( 6x \).
- The derivative of \( \tan^{-1}(x) \) is \( \frac{1}{1+x^2} \).

So:

\[
\frac{du}{dx} = 6x + \frac{1}{1+x^2}
\]

### Step 3: Combine the Results

Now substitute \( \frac{du}{dx} \) into the chain rule expression:

\[
\frac{dy}{dx} = 4 \left(3x^2 + \tan^{-1}(x)\right)^3 \cdot \left(6x + \frac{1}{1+x^2}\right)
\]

This is the derivative of the given function. According to the problem instructions, we are not required to simplify further.

Would you like more details on any of the steps or explanations? Here are some related questions to expand your understanding:

1. How does the chain rule apply to composite functions?
2. Why is the derivative of \( \tan^{-1}(x) \) equal to \( \frac{1}{1+x^2} \)?
3. What is the general formula for the derivative of \( f(x)^n \)?
4. Could the product rule or quotient rule apply in any part of this problem?
5. What would happen if we simplified the final result?

**Tip:** When applying the chain rule, always identify the outer function and inner function first to avoid confusion.

Find the derivative of y = (3x^2 + tan⁻¹(x))^4. Do NOT simplify further once all needed derivatives have been taken.

This problem involves finding the derivative of y = (3x^2 + tan⁻¹(x))^4 using the chain rule. The solution covers the application of differentiation techniques such as the chain rule and the power rule. It is suitable for advanced high school or university-level calculus students.

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