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The image shows the derivative of the natural logarithm function \( \ln(u) \) with respect to \( u \) and \( x \). Let's break it down:

1. The first equation is:

\[
\frac{d}{du} \ln(u) = \frac{1}{u}
\]

This is the derivative of the natural logarithm function. For any function \( u \), the derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \).

2. The second equation appears to involve the chain rule for differentiation:

\[
\frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{u} \frac{du}{dx}
\]

This is the derivative of \( \ln(u) \) with respect to \( x \), where \( u \) is a function of \( x \) (denoted as \( u(x) \)). The chain rule is applied here, so we first differentiate \( \ln(u) \) with respect to \( u \), giving \( \frac{1}{u} \), and then multiply by \( \frac{du}{dx} \), the derivative of \( u \) with respect to \( x \).

### Summary:
- The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \).
- If \( u \) is a function of \( x \), the derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).

Would you like more details on the differentiation rules or how the chain rule works?

Here are some related questions:
1. What is the derivative of \( \ln(x^2) \) with respect to \( x \)?
2. How would you apply the chain rule for \( \ln(u(x) + v(x)) \)?
3. What is the second derivative of \( \ln(u) \) with respect to \( x \)?
4. How does the chain rule apply to other composite functions?
5. Can you find the derivative of \( \ln(e^{2x}) \)?

**Tip:** When differentiating logarithmic functions, always remember that the chain rule applies if the logarithmic argument is a function of another variable.

Explore the derivative of the natural logarithm function ln(u) with respect to both u and x, applying the chain rule of differentiation. Learn how the derivative is broken down and how to apply the chain rule when differentiating composite functions. This problem is suitable for students in undergraduate-level calculus.

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