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The image contains the following mathematical expression related to differentiation and logarithmic functions:

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

### Explanation:
1. The derivative of the natural logarithm of a function \( u \) with respect to \( u \) is given by:
   \[
   \frac{d}{du} \ln u = \frac{1}{u}
   \]
   This is a fundamental result in calculus.

2. Next, the expression shows how to apply this derivative when \( u \) is a function of \( x \). The derivative of \( \ln u(x) \) with respect to \( x \) involves the chain rule:
   \[
   \frac{d}{dx} \ln u(x) = \frac{1}{u(x)} \cdot \frac{du}{dx}
   \]
   This simplifies to:
   \[
   \frac{\frac{du}{dx}}{u(x)}
   \]
   which shows how the derivative of a composite function \( \ln u(x) \) is computed.

Would you like a more detailed explanation or do you have any specific questions?

Here are 5 related questions for further exploration:
1. How do you apply the chain rule to more complex functions involving logarithms?
2. What is the general formula for the derivative of \( \ln f(x) \), where \( f(x) \) is a function of \( x \)?
3. How would the derivative change if you were differentiating \( \ln(u(x)^n) \)?
4. Can you derive the second derivative of \( \ln u(x) \)?
5. How do you integrate \( \frac{1}{u} \cdot \frac{du}{dx} \) to find \( u(x) \)?

### Tip:
When differentiating a composite function, always check if the chain rule is needed to handle nested functions correctly.

Differentiate the natural logarithm of a function u with respect to u and then with respect to x, where u is a function of x.

Explore how to differentiate the natural logarithm of a function u(x) using the chain rule. This calculus problem breaks down the steps for finding the derivative of ln(u) with respect to both u and x, suitable for advanced high school or college students.

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