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To differentiate \( f(x) = \ln(7x^2 - 17) \), we can apply the chain rule. Here's the step-by-step process:

1. **Identify the Outer and Inner Functions**:
   - The outer function is the natural logarithm \( \ln(u) \), where \( u = 7x^2 - 17 \).
   - The inner function is \( u = 7x^2 - 17 \).

2. **Differentiate the Outer Function**:
   - The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \).

3. **Differentiate the Inner Function**:
   - The derivative of \( u = 7x^2 - 17 \) with respect to \( x \) is \( 14x \).

4. **Apply the Chain Rule**:
   \[
   f'(x) = \frac{1}{7x^2 - 17} \cdot 14x
   \]

5. **Simplify the Result**:
   \[
   f'(x) = \frac{14x}{7x^2 - 17}
   \]

So, the derivative of \( f(x) = \ln(7x^2 - 17) \) is:
\[
f'(x) = \frac{14x}{7x^2 - 17}
\]

Would you like a deeper explanation of any steps, or do you have any questions?

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### Related Questions
1. How would you differentiate \( f(x) = \ln(ax^2 + bx + c) \) for general constants \( a \), \( b \), and \( c \)?
2. What are some real-world applications of logarithmic differentiation?
3. How does the chain rule apply to functions with multiple nested components?
4. What is the derivative of \( g(x) = e^{7x^2 - 17} \)?
5. How would you handle the differentiation if the argument inside the logarithm had a variable base?

### Tip
When differentiating a composite function like \( \ln(g(x)) \), always start by identifying the outer and inner functions to apply the chain rule effectively.

Learn the step-by-step process of differentiating f(x) = ln(7x^2 - 17) using the chain rule. This guide covers the outer and inner function approach, making it ideal for college-level calculus students.

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