https://file.aiquickdraw.com/mathbot/file/172949040994437hu3d6l.jpg

Let's walk through the problem step-by-step to find the derivative of \( q(x) = \log_8(6x^3 + 3x + 1) \).

### Step 1: Rewrite the logarithmic function using the change of base formula
The derivative of a logarithmic function with any base \(a\) can be expressed using the natural logarithm:
\[
\log_a(f(x)) = \frac{\ln(f(x))}{\ln(a)}
\]
Thus, we can rewrite:
\[
q(x) = \frac{\ln(6x^3 + 3x + 1)}{\ln(8)}
\]

### Step 2: Differentiate using the chain rule
The derivative \(q'(x)\) involves two parts: 
1. Differentiating the numerator: \( \ln(6x^3 + 3x + 1) \).
2. Accounting for the constant denominator: \( \ln(8) \).

#### Part 1: Derivative of \( \ln(6x^3 + 3x + 1) \)
Using the chain rule:
\[
\frac{d}{dx} \left( \ln(6x^3 + 3x + 1) \right) = \frac{18x^2 + 3}{6x^3 + 3x + 1}
\]

#### Part 2: Incorporate the constant \( \frac{1}{\ln(8)} \)
Since \( \ln(8) \) is a constant, the full derivative becomes:
\[
q'(x) = \frac{1}{\ln(8)} \cdot \frac{18x^2 + 3}{6x^3 + 3x + 1}
\]

### Step 3: Final expression
Thus, the derivative is:
\[
q'(x) = \frac{18x^2 + 3}{\ln(8) \cdot (6x^3 + 3x + 1)}
\]

### Step 4: Match the correct answer
From the options provided in the image, the correct answer is:
\[
\boxed{\frac{18x^2 + 3}{\ln(8) \cdot (6x^3 + 3x + 1)}}
\]

This corresponds to the **second-to-last option** in the list.

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Do you have any questions or need further clarifications?

### Five related questions:
1. What is the general rule for differentiating logarithms with arbitrary bases?
2. How do we apply the chain rule in more complicated functions?
3. Why do we need the change of base formula for logarithmic differentiation?
4. What happens to the derivative if the base of the logarithm is \(e\)?
5. How can logarithmic differentiation help in solving product or quotient rules?

### Tip:
When differentiating logarithmic functions, always remember that switching to the natural logarithm can simplify the derivative by introducing constants, making it easier to apply the chain rule.

What is q'(x) when q(x) = log_8(6x^3 + 3x + 1)?

Learn how to compute the derivative of q(x) = log_8(6x^3 + 3x + 1) step-by-step using the chain rule and logarithmic differentiation. Suitable for calculus students.

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