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To differentiate the given function \( f(x) = \log_{10}(x^3 + 2) \), we will apply the chain rule and the derivative of the logarithmic function. Here are the steps:

1. **Recall the derivative of \( \log_b(u) \)**:  
   If \( f(x) = \log_b(u(x)) \), the derivative is given by:
   \[
   f'(x) = \frac{1}{u(x) \ln(b)} \cdot u'(x)
   \]
   where \( u(x) \) is the inner function, and \( b \) is the base of the logarithm (in this case, 10).

2. **Identify \( u(x) \)**:  
   For this function, \( u(x) = x^3 + 2 \).

3. **Differentiate \( u(x) \)**:  
   The derivative of \( u(x) = x^3 + 2 \) is:
   \[
   u'(x) = 3x^2
   \]

4. **Apply the chain rule**:  
   Using the formula for the derivative of a logarithmic function, we get:
   \[
   f'(x) = \frac{1}{(x^3 + 2) \ln(10)} \cdot 3x^2
   \]

Thus, the derivative of the function is:
\[
f'(x) = \frac{3x^2}{(x^3 + 2) \ln(10)}
\]

Let me know if you need further clarification! Would you like more details on any step?

### Related Questions:
1. How would the derivative change if the logarithm had a different base?
2. What happens if the inner function was more complex, such as \( x^4 + 3x^2 + 1 \)?
3. How does the chain rule apply to other types of functions like exponentials?
4. Can you explain the role of \( \ln(10) \) in this derivative?
5. How would this problem change if the logarithm was natural, \( \ln(x^3 + 2) \)?

### Tip:
Always make sure to identify the inner function clearly when using the chain rule for differentiation!

Differentiate the function f(x) = log_10(x^3 + 2)

Find the derivative of the function f(x) = log_10(x^3 + 2) with a step-by-step solution. This calculus problem uses logarithmic differentiation and the chain rule to simplify the process. Ideal for students studying calculus at the high school or college level.

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