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We are tasked with finding the derivative of the function:

\[
y = \left( \tan^{-1}(4x) \right)^2
\]

### Step-by-step solution:

1. **Apply the chain rule**:
   To differentiate this expression, we treat the outer function as \( u^2 \) where \( u = \tan^{-1}(4x) \). So the derivative of \( y = u^2 \) with respect to \( u \) is:
   \[
   \frac{d}{dx}[u^2] = 2u \cdot \frac{du}{dx}
   \]
   Therefore:
   \[
   y' = 2 \cdot \tan^{-1}(4x) \cdot \frac{d}{dx}\left[ \tan^{-1}(4x) \right]
   \]

2. **Differentiate \( \tan^{-1}(4x) \)**:
   Use the chain rule again to differentiate \( \tan^{-1}(4x) \). The derivative of \( \tan^{-1}(u) \) is \( \frac{1}{1 + u^2} \), and since \( u = 4x \), we also multiply by the derivative of \( 4x \) (which is 4):
   \[
   \frac{d}{dx}\left[ \tan^{-1}(4x) \right] = \frac{4}{1 + (4x)^2} = \frac{4}{1 + 16x^2}
   \]

3. **Combine the results**:
   Now substitute this back into the expression for \( y' \):
   \[
   y' = 2 \cdot \tan^{-1}(4x) \cdot \frac{4}{1 + 16x^2}
   \]
   Simplifying:
   \[
   y' = \frac{8 \cdot \tan^{-1}(4x)}{1 + 16x^2}
   \]

Thus, the derivative of the given function is:

\[
y' = \frac{8 \cdot \tan^{-1}(4x)}{1 + 16x^2}
\]

Would you like more details or clarification on any step?

Here are five related questions to expand your understanding:

1. What are the properties of the inverse tangent function, \(\tan^{-1}(x)\)?
2. How does the chain rule work for compositions of functions?
3. Can you explain why the derivative of \(\tan^{-1}(x)\) is \(\frac{1}{1 + x^2}\)?
4. How would the derivative change if the argument inside the inverse tangent were a more complex function of \(x\)?
5. Can we apply a similar method to differentiate powers of other inverse trigonometric functions?

**Tip:** When differentiating functions that involve inverse trigonometric expressions, always pay close attention to the chain rule and any inner functions you need to differentiate.

Find the derivative of the function: y = (tan^(-1)(4x))^2

Learn how to differentiate the function y = (tan^(-1)(4x))^2 with a detailed step-by-step solution. This problem involves the chain rule and the derivative of inverse trigonometric functions, suitable for undergraduate calculus students.

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