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The expression you're asked to differentiate is:

\[
D_x \left( \ln(6x^2 + 3) \right)
\]

### Step-by-Step Solution:

1. **Recall the derivative of the natural logarithm**:
   The derivative of \(\ln(u)\), where \(u = f(x)\), is given by:
   \[
   \frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx}
   \]
   
2. **Identify \(u\)**:
   Here, \(u = 6x^2 + 3\).

3. **Differentiate \(u = 6x^2 + 3\)**:
   \[
   \frac{du}{dx} = \frac{d}{dx}(6x^2 + 3) = 12x
   \]

4. **Apply the chain rule**:
   Now, applying the chain rule to differentiate \(\ln(6x^2 + 3)\):
   \[
   \frac{d}{dx} \ln(6x^2 + 3) = \frac{1}{6x^2 + 3} \cdot 12x
   \]

5. **Simplify**:
   \[
   \frac{12x}{6x^2 + 3}
   \]

So, the derivative of \(\ln(6x^2 + 3)\) with respect to \(x\) is:

\[
\frac{12x}{6x^2 + 3}
\]

Would you like further details or have any questions?

### Related Questions:
1. What is the derivative of \(\ln(x^2 + 1)\)?
2. How would you differentiate \( \ln(3x^2 - 5x + 4)\)?
3. What is the second derivative of \(\ln(6x^2 + 3)\)?
4. Can you differentiate \( \ln(ax^2 + bx + c) \) in general form?
5. How is the derivative of \(\ln(f(x))\) applied in optimization problems?

### Tip:
When differentiating logarithmic functions, always remember to apply the chain rule, especially when the argument is more than just \(x\).

Learn how to differentiate the natural logarithmic function ln(6x^2 + 3) using the chain rule. This detailed step-by-step solution covers differentiation techniques, including logarithmic differentiation and the chain rule. Suitable for Grade 11-12 students and early calculus learners.

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