The derivative of log x with repect to `(1)/(x)` is

Options

A. `-(1)/(x^(3))`
B. `-(1)/(x)`
C. `-x`
D. `(1)/(x)`

Answer

To find the derivative of \( \log x \) with respect to \( \frac{1}{x} \), we can follow these steps:

### Step 1: Define the variables
Let:
- \( u = \log x \)
- \( v = \frac{1}{x} \)

### Step 2: Use the chain rule
The derivative of \( u \) with respect to \( v \) can be expressed using the chain rule:
\[ \frac{du}{dv} = \frac{du}{dx} \cdot \frac{dx}{dv} \]

### Step 3: Find \( \frac{du}{dx} \)
To find \( \frac{du}{dx} \):
\[ u = \log x \implies \frac{du}{dx} = \frac{1}{x} \]

### Step 4: Find \( \frac{dv}{dx} \)
To find \( \frac{dv}{dx} \):
\[ v = \frac{1}{x} \implies \frac{dv}{dx} = -\frac{1}{x^2} \]

### Step 5: Substitute into the chain rule
Now substitute \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) into the chain rule formula:
\[ \frac{du}{dv} = \frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{\frac{1}{x}}{-\frac{1}{x^2}} \]

### Step 6: Simplify the expression
Simplifying the expression:
\[ \frac{du}{dv} = \frac{1}{x} \cdot \left(-x^2\right) = -x \]

### Conclusion
Thus, the derivative of \( \log x \) with respect to \( \frac{1}{x} \) is:
\[ \frac{d(\log x)}{d(\frac{1}{x})} = -x \]

### Final Answer
The final answer is \( -x \).

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