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The problem asks for the derivative of the function \( y = \log \left|\sec(e^{x^2})\right| \) with respect to \( x \).

### Step-by-Step Solution:
1. **Identify the function**:
   \[
   y = \log \left|\sec(e^{x^2})\right|
   \]

2. **Differentiate \( y \) with respect to \( x \)**:
   To find \(\frac{dy}{dx}\), apply the chain rule. First, differentiate the logarithm function.

   The derivative of \( \log |u| \) with respect to \( u \) is:
   \[
   \frac{d}{du} \log |u| = \frac{1}{u} \cdot \frac{du}{dx}
   \]
   where \( u = \sec(e^{x^2}) \).

   Therefore, the derivative becomes:
   \[
   \frac{dy}{dx} = \frac{1}{\sec(e^{x^2})} \cdot \frac{d}{dx} \sec(e^{x^2})
   \]

3. **Differentiate the secant function**:
   The derivative of \( \sec(u) \) with respect to \( u \) is:
   \[
   \frac{d}{du} \sec(u) = \sec(u) \tan(u)
   \]
   where \( u = e^{x^2} \).

   Thus:
   \[
   \frac{d}{dx} \sec(e^{x^2}) = \sec(e^{x^2}) \cdot \tan(e^{x^2}) \cdot \frac{d}{dx} e^{x^2}
   \]

4. **Differentiate the exponential function**:
   The derivative of \( e^{x^2} \) with respect to \( x \) is:
   \[
   \frac{d}{dx} e^{x^2} = e^{x^2} \cdot 2x
   \]

5. **Combine all the derivatives**:
   Substitute the derivative of \( e^{x^2} \) into the expression for \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx} = \frac{1}{\sec(e^{x^2})} \cdot \sec(e^{x^2}) \tan(e^{x^2}) \cdot e^{x^2} \cdot 2x
   \]

   Simplifying:
   \[
   \frac{dy}{dx} = \tan(e^{x^2}) \cdot e^{x^2} \cdot 2x
   \]

### Final Answer:
\[
\frac{dy}{dx} = 2x \cdot e^{x^2} \cdot \tan(e^{x^2})
\]

Would you like further clarification on any part of this process, or do you have additional questions?

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**Relative Questions:**
1. What is the derivative of \( \sec(x) \) with respect to \( x \)?
2. How does the chain rule apply to functions composed of logarithmic and trigonometric functions?
3. What are the general rules for differentiating logarithmic functions?
4. How would the derivative change if the argument inside the secant was a linear function instead of an exponential?
5. Can you derive the expression using logarithmic differentiation?

**Tip:** When dealing with derivatives of composite functions, always apply the chain rule carefully to ensure each layer of the function is correctly differentiated.

Learn how to find the derivative of the function y = log |sec(e^{x^2})| using the chain rule and differentiation of logarithmic and trigonometric functions. Detailed step-by-step solution provided.

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