https://file.aiquickdraw.com/mathbot/file/1729487380656z0njf2kb.jpg

To solve the problem, we need to find the derivative of the function:

\[
f(x) = \log_3\left(\frac{3x^2}{\sqrt{x}}\right)
\]

### Step 1: Simplify the expression inside the logarithm
We can rewrite the argument inside the logarithm to simplify the differentiation process:

\[
\frac{3x^2}{\sqrt{x}} = 3x^{2 - \frac{1}{2}} = 3x^{\frac{3}{2}}
\]

So, the function becomes:

\[
f(x) = \log_3(3x^{\frac{3}{2}})
\]

### Step 2: Use the properties of logarithms
We can apply the logarithmic property \(\log_b(mn) = \log_b(m) + \log_b(n)\):

\[
f(x) = \log_3(3) + \log_3(x^{\frac{3}{2}})
\]

Since \(\log_3(3) = 1\), this simplifies to:

\[
f(x) = 1 + \frac{3}{2} \log_3(x)
\]

### Step 3: Differentiate the function
Now, differentiate \(f(x)\) term by term.

- The derivative of \(1\) is \(0\).
- The derivative of \(\frac{3}{2} \log_3(x)\) can be found using the derivative formula for logarithms with a base \(b\), which is:

\[
\frac{d}{dx} \log_b(x) = \frac{1}{x \ln(b)}
\]

Thus, the derivative of \(\frac{3}{2} \log_3(x)\) is:

\[
\frac{3}{2} \cdot \frac{1}{x \ln(3)} = \frac{3}{2x \ln(3)}
\]

### Final Answer:
The derivative of the function is:

\[
f'(x) = \frac{3}{2x \ln(3)}
\]

Looking at the provided options, the correct answer is:

\[
\boxed{\frac{3}{2x \ln(3)}}
\]

Would you like more details or have any other questions? Here are some related questions to explore:

1. What is the chain rule, and how does it apply to logarithmic derivatives?
2. How does the base of the logarithm affect the derivative?
3. Can you explain the properties of logarithms used in simplifying the function?
4. How do we differentiate a logarithm with respect to a function of \(x\)?
5. What are the rules for differentiating power functions?

**Tip:** Always simplify the logarithmic expression as much as possible before differentiating to make the process easier!

Find the derivative of the function f(x) = log_3((3x^2) / sqrt(x)).

This article provides a step-by-step solution to finding the derivative of f(x) = log_3((3x^2) / sqrt(x)). It explores key concepts such as logarithmic differentiation, logarithmic properties, and how to apply them effectively. Suitable for high school students in Grades 11-12.

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