To find the derivative of \( f(x) = \ln(3x^2 - 5x) \), we will use the chain rule.

### Steps:
1. **Outer function**: The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \).
2. **Inner function**: The inner function is \( u(x) = 3x^2 - 5x \). Its derivative with respect to \( x \) is \( u'(x) = 6x - 5 \).

Now apply the chain rule:
\[
\frac{d}{dx} \left[ \ln(3x^2 - 5x) \right] = \frac{1}{3x^2 - 5x} \cdot (6x - 5)
\]

So, the derivative is:
\[
\frac{6x - 5}{3x^2 - 5x}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the derivative change if the function was \( \ln(3x^2 + 5x) \)?
2. What is the second derivative of \( \ln(3x^2 - 5x) \)?
3. How do you find the derivative of other logarithmic functions like \( \ln(2x + 1) \)?
4. How do logarithmic differentiation techniques help with more complex functions?
5. Can you derive the general form of the derivative of \( \ln(ax^2 + bx + c) \)?

**Tip**: When dealing with the derivative of logarithmic functions, always apply the chain rule by first differentiating the logarithmic part and then the inner function.

Find the derivative of ln(3x^2 - 5x) using the chain rule. This calculus problem breaks down the process, starting with differentiating the logarithmic function and applying the chain rule to the inner function. Suitable for students in Grades 11-12 and early college.

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