https://file.aiquickdraw.com/mathbot/file/17294553753641gcl20b3.jpg

To find the derivative of the function \( f(x) = \ln(2x^4 + x^2 - 3) \), we will use the chain rule and the derivative of a logarithmic function.

### Step-by-step Solution:

1. **Recall the derivative of \( \ln(u) \)**: 
   \[
   \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx}
   \]
   where \( u = 2x^4 + x^2 - 3 \).

2. **Differentiate \( u = 2x^4 + x^2 - 3 \)**:
   \[
   \frac{du}{dx} = \frac{d}{dx}(2x^4 + x^2 - 3) = 8x^3 + 2x
   \]

3. **Apply the chain rule**:
   Using the chain rule for \( f(x) = \ln(u) \), we get:
   \[
   f'(x) = \frac{1}{u} \cdot \frac{du}{dx}
   \]
   Substituting \( u = 2x^4 + x^2 - 3 \) and \( \frac{du}{dx} = 8x^3 + 2x \), we have:
   \[
   f'(x) = \frac{1}{2x^4 + x^2 - 3} \cdot (8x^3 + 2x)
   \]

This is the derivative of the given function.

### Now, checking the options:
- The correct answer is:
  \[
  \frac{8x^3 + 2x}{2x^4 + x^2 - 3}
  \]
  which corresponds to the third option in the image.

Would you like more details on any part of the process or have any further questions?

Here are 5 related questions you might explore:
1. How does the chain rule apply to other types of functions like exponential or trigonometric functions?
2. What are some common mistakes when differentiating logarithmic functions?
3. How can implicit differentiation be used for more complex logarithmic functions?
4. What is the relationship between natural logs and exponential growth in real-world applications?
5. How can we differentiate more complicated compositions involving products of logarithmic functions?

**Tip:** When differentiating complex expressions, always simplify terms as much as possible before applying rules like the chain rule or product rule.

Find the derivative of the function f(x) = ln(2x^4 + x^2 - 3).

Learn how to differentiate the logarithmic function f(x) = ln(2x^4 + x^2 - 3) using the chain rule. This calculus problem covers key concepts such as logarithmic differentiation and the application of the chain rule, suitable for advanced high school or undergraduate students.

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