https://file.aiquickdraw.com/mathbot/file/172948685757949l7qrku.jpg

To find the first derivative of \( f(x) = \log_2(6x^3 + 2x + 6) \), we need to apply the chain rule. Let's go through the process step by step.

### Step 1: Understand the derivative formula

The derivative of \( \log_a(u) \) with respect to \( x \) is:
\[
\frac{d}{dx}[\log_a(u)] = \frac{1}{u \ln(a)} \cdot \frac{du}{dx}
\]

where \( a = 2 \) and \( u = 6x^3 + 2x + 6 \).

### Step 2: Differentiate the function

1. Identify \( u(x) \):
   \[
   u = 6x^3 + 2x + 6
   \]
   
2. Find \( \frac{du}{dx} \):
   \[
   \frac{du}{dx} = 18x^2 + 2
   \]

3. Plug these into the derivative formula:
   \[
   f'(x) = \frac{1}{(6x^3 + 2x + 6) \ln(2)} \cdot (18x^2 + 2)
   \]

### Step 3: Simplify the expression

\[
f'(x) = \frac{18x^2 + 2}{(6x^3 + 2x + 6) \ln(2)}
\]

### Step 4: Compare with the given options

From the options provided in the image, the correct answer matches:
\[
\frac{18x^2 + 2}{(\ln 2)(6x^3 + 2x + 6)}
\]

Therefore, the correct answer is:
\[
\frac{18x^2 + 2}{(\ln 2)(6x^3 + 2x + 6)}
\]

### Do you want details or have any questions?
Here are 5 related questions to deepen your understanding:

1. How do you find the derivative of a logarithmic function with a base other than \( e \)?
2. Can you explain why the chain rule is necessary in this problem?
3. How would the process change if the base of the logarithm were \( e \) (natural log)?
4. What is the derivative of \( \log_a(x) \) where \( a \) is any constant?
5. How would you apply the same process to find the second derivative of \( f(x) \)?

**Tip:** When differentiating logarithmic functions, always remember to handle the base correctly by including \( \ln(a) \) when the base is not \( e \).

What is the first derivative of f(x) = log_2(6x^3 + 2x + 6)?

Learn how to differentiate the function f(x) = log_2(6x^3 + 2x + 6) using the chain rule. This solution involves logarithmic differentiation and is suitable for undergraduate-level calculus students.

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