Differentiate the function. If​ possible, first use properties of logarithms to simplify the given function.
​f(x)equalsln left parenthesis StartFraction x cubed minus 4 Over x EndFraction right parenthesis

To differentiate the function \( f(x) = \ln\left(\frac{x^3 - 4}{x}\right) \), we can first simplify it using the properties of logarithms. Specifically, we use the property:

\[
\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)
\]

### Step 1: Simplify using logarithmic properties

Apply the property to the given function:

\[
f(x) = \ln(x^3 - 4) - \ln(x)
\]

### Step 2: Differentiate the simplified function

Now we differentiate each term separately:

1. For the term \( \ln(x^3 - 4) \), we apply the chain rule:

\[
\frac{d}{dx} \left( \ln(x^3 - 4) \right) = \frac{1}{x^3 - 4} \cdot \frac{d}{dx} \left( x^3 - 4 \right) = \frac{1}{x^3 - 4} \cdot 3x^2 = \frac{3x^2}{x^3 - 4}
\]

2. For the term \( \ln(x) \), the derivative is:

\[
\frac{d}{dx} \left( \ln(x) \right) = \frac{1}{x}
\]

### Step 3: Combine the results

The derivative of \( f(x) \) is:

\[
f'(x) = \frac{3x^2}{x^3 - 4} - \frac{1}{x}
\]

This is the derivative of the given function.

Would you like further clarification or more details on this? Here are some related questions:

1. How do logarithmic properties help in differentiation?
2. Can we differentiate without using logarithmic simplification?
3. What is the chain rule in calculus?
4. How do you simplify complex fractions after differentiation?
5. Can this method be applied to other types of logarithmic functions?

**Tip:** Simplifying a function before differentiation often leads to easier and faster results.

Differentiate the function. If possible, first use properties of logarithms to simplify the given function. f(x) = ln((x^3 - 4) / x)

Learn how to differentiate the logarithmic function f(x) = ln((x^3 - 4) / x) by simplifying with logarithmic properties and applying the chain rule. This detailed solution walks through the process step-by-step, using key differentiation techniques. Suitable for advanced high school and early college students.

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