What is the differentiation of `log_(x)x` with respect to ln x?

Options

A. 0
B. 1
C. 1/x
D. x

Explanation

To differentiate \( \log_{x} x \) with respect to \( \ln x \), we can follow these steps:

### Step 1: Rewrite the logarithm
We can rewrite \( \log_{x} x \) using the change of base formula:
\[ \log_{x} x = \frac{\ln x}{\ln x} = 1. \]

### Step 2: Differentiate with respect to \( \ln x \)
Since \( \log_{x} x = 1 \) is a constant, its derivative with respect to any variable is 0:
\[ \frac{d}{d(\ln x)} (\log_{x} x) = \frac{d}{d(\ln x)} (1) = 0. \]

### Step 3: Conclusion
Thus, the differentiation of \( \log_{x} x \) with respect to \( \ln x \) is:
\[ \frac{d}{d(\ln x)} (\log_{x} x) = 0. \]

### Final Answer:
\[ \frac{d}{d(\ln x)} (\log_{x} x) = 0. \]

---