We are given the function:

$f(x) = 9 \ln(9 + 5x^2)$

To find $f'(x)$, we need to differentiate this function using the chain rule.

Step 1: Differentiate the outer function

The outer function is $9 \ln(u)$, where $u = 9 + 5x^2$. The derivative of $\ln(u)$ is $\frac{1}{u}$, so:

$\frac{d}{dx} [9 \ln(u)] = 9 \cdot \frac{1}{u} \cdot \frac{du}{dx}$

Step 2: Differentiate the inner function

Now, let's differentiate the inner function $u = 9 + 5x^2$. The derivative of $u$ with respect to $x$ is:

$\frac{du}{dx} = \frac{d}{dx} [9 + 5x^2] = 10x$

Step 3: Apply the chain rule

Substitute back the derivative of $u$ into the derivative of the outer function:

$f'(x) = 9 \cdot \frac{1}{9 + 5x^2} \cdot 10x$

Step 4: Simplify

Simplifying the expression:

$f'(x) = \frac{90x}{9 + 5x^2}$

So, the derivative of $f(x)$ is:

$f'(x) = \frac{90x}{9 + 5x^2}$

Would you like more details on any part of this solution?

Here are 5 related questions to explore:

1. How do you apply the chain rule in differentiation?
2. What is the derivative of the natural logarithm function?
3. How do you handle the product of constants when differentiating?
4. Can you explain the importance of simplifying derivatives?
5. How would the derivative change if the logarithmic function were base 10?

Tip: Always remember to differentiate both the outer and inner functions when using the chain rule.