Differentiate.

 f left parenthesis x right parenthesis equals Bold ln left parenthesis 8 x right parenthesisf(x)=**ln**(8x)




To differentiate \( f(x) = \ln(8x) \), we will apply the chain rule.

### Step-by-step differentiation:

1. Recall that the derivative of \( \ln(u) \) is \( \frac{1}{u} \), where \( u \) is a function of \( x \). 
   
2. For \( f(x) = \ln(8x) \), \( u = 8x \). Now, the derivative of \( u \) with respect to \( x \) is:
   \[
   \frac{d}{dx}(8x) = 8
   \]

3. Applying the chain rule, the derivative of \( f(x) \) is:
   \[
   f'(x) = \frac{1}{8x} \cdot 8
   \]

4. Simplifying:
   \[
   f'(x) = \frac{8}{8x} = \frac{1}{x}
   \]

Thus, the derivative of \( f(x) = \ln(8x) \) is:
\[
f'(x) = \frac{1}{x}
\]

Would you like further details or clarification on this process?

Here are 5 related questions to explore further:

1. How do you differentiate other logarithmic functions like \( \ln(ax + b) \)?
2. How do you apply the chain rule to more complex functions?
3. What is the derivative of exponential functions like \( e^{x} \)?
4. Can we differentiate logarithms with different bases, such as \( \log_b(x) \)?
5. How does the product rule interact with logarithmic differentiation?

**Tip**: When differentiating logarithmic functions, always apply the chain rule if the argument of the logarithm is more complex than just \( x \).

This page provides a detailed solution for differentiating the function f(x) = ln(8x) using the chain rule. Learn how to apply the logarithmic differentiation technique, understand key calculus concepts, and see the step-by-step solution. Ideal for students in Grades 11-12 or college level.

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