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 $x$.