We are given a function:

$f(x) = \ln \left[ x^2 (x+3)^7 (x^2 + 6)^5 \right]$

To find $f'(x)$, we can apply the chain rule in combination with the product rule. However, logarithmic differentiation can make this process simpler. Using the properties of logarithms, we can rewrite the function as:

$f(x) = \ln(x^2) + \ln\left((x+3)^7\right) + \ln\left((x^2 + 6)^5\right)$

This simplifies further using logarithmic identities:

$f(x) = 2\ln(x) + 7\ln(x+3) + 5\ln(x^2 + 6)$

Now, we differentiate each term separately:

1. $\frac{d}{dx} \left( 2 \ln(x) \right) = \frac{2}{x}$
2. $\frac{d}{dx} \left( 7 \ln(x+3) \right) = \frac{7}{x+3}$
3. $\frac{d}{dx} \left( 5 \ln(x^2 + 6) \right) = 5 \cdot \frac{d}{dx} \left( \ln(x^2 + 6) \right) = 5 \cdot \frac{2x}{x^2 + 6} = \frac{10x}{x^2 + 6}$

Now, we can combine these results to get:

$f'(x) = \frac{2}{x} + \frac{7}{x+3} + \frac{10x}{x^2 + 6}$

This is the derivative of the given function.

Let me know if you have any questions, or if you'd like further details on any step!

Related questions:

1. How would you apply the product rule directly without logarithmic differentiation?
2. How does the chain rule apply when differentiating the logarithmic terms?
3. What other functions would benefit from logarithmic differentiation?
4. How would you handle a similar problem if the base of the logarithm were different (e.g., $\log_{10}$ instead of $\ln$)?
5. How does the logarithmic identity $\ln(ab) = \ln(a) + \ln(b)$ simplify the process?

Tip:

When differentiating logarithmic functions involving products or powers, always consider using logarithmic properties to simplify the expression before differentiating!