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

\[
f(x) = \log_6 \left( \sqrt[3]{\frac{10 - x}{-2x - 5}} \right)
\]

### Step 1: Simplifying using logarithmic properties
We will use the following logarithmic properties:

- \( \log_b(a^c) = c \cdot \log_b(a) \)
- The derivative of \( \log_b(u) \) is \( \frac{1}{u \ln(b)} \cdot u' \)

Apply the first property to simplify the function:

\[
f(x) = \frac{1}{3} \log_6 \left( \frac{10 - x}{-2x - 5} \right)
\]

### Step 2: Differentiate the simplified form
Now we differentiate \( f(x) \). First, we apply the chain rule to handle the factor \( \frac{1}{3} \):

\[
f'(x) = \frac{1}{3} \cdot \frac{d}{dx} \left( \log_6 \left( \frac{10 - x}{-2x - 5} \right) \right)
\]

The derivative of \( \log_b(u) \) is \( \frac{1}{u \ln(b)} \cdot u' \), so:

\[
f'(x) = \frac{1}{3} \cdot \frac{1}{\frac{10 - x}{-2x - 5} \ln(6)} \cdot \frac{d}{dx} \left( \frac{10 - x}{-2x - 5} \right)
\]

### Step 3: Differentiate the inner function
We now need to find the derivative of:

\[
\frac{10 - x}{-2x - 5}
\]

Using the quotient rule, \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v u' - u v'}{v^2} \), where:

- \( u = 10 - x \), so \( u' = -1 \)
- \( v = -2x - 5 \), so \( v' = -2 \)

Applying the quotient rule:

\[
\frac{d}{dx} \left( \frac{10 - x}{-2x - 5} \right) = \frac{(-2x - 5)(-1) - (10 - x)(-2)}{(-2x - 5)^2}
\]

Simplify the numerator:

\[
= \frac{(2x + 5) - (20 - 2x)}{(-2x - 5)^2}
= \frac{2x + 5 - 20 + 2x}{(-2x - 5)^2}
= \frac{4x - 15}{(-2x - 5)^2}
\]

### Step 4: Combine everything
Now substitute this back into the expression for \( f'(x) \):

\[
f'(x) = \frac{1}{3} \cdot \frac{1}{\frac{10 - x}{-2x - 5} \ln(6)} \cdot \frac{4x - 15}{(-2x - 5)^2}
\]

Simplify:

\[
f'(x) = \frac{1}{3 \ln(6)} \cdot \frac{4x - 15}{(10 - x)(-2x - 5)}
\]

Thus, the derivative of the function is:

\[
f'(x) = \frac{(4x - 15)}{3 \ln(6) \cdot (10 - x)(-2x - 5)}
\]

Would you like further clarification or more details on any part of this process?

Here are 5 related questions to explore further:

1. How do logarithmic properties simplify differentiation?
2. What are the steps in applying the chain rule with logarithmic functions?
3. How does the quotient rule apply to rational functions?
4. What is the significance of the natural logarithm \( \ln(b) \) in logarithmic differentiation?
5. How do we simplify complex fractions when solving derivatives?

**Tip:** When applying the chain rule, always differentiate the outer function first and then multiply by the derivative of the inner function.

Find the derivative of f(x) = log_6(∛((10 - x)/(-2x - 5))) using the properties of logarithms.

Learn how to find the derivative of f(x) = log_6(∛((10 - x)/(-2x - 5))) using logarithmic properties. This step-by-step solution covers the Chain Rule, Quotient Rule, and properties of logarithms, suitable for advanced high school students.

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