If ` y = x^(1//x)` , then the derivative of y is

Options

A. `(y)/(x^(2)) (1 -log x)`
B. `(y)/(x^(2)) (1 + log x)`
C. `-(y)/(x^(2))(1 -log x)`
D. None of these

Explanation

To find the derivative of the function \( y = x^{\frac{1}{x}} \), we will follow these steps:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of both sides to simplify the differentiation process.

\[ \ln y = \ln\left(x^{\frac{1}{x}}\right) \]

### Step 2: Simplify using logarithmic properties
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can simplify the right-hand side:

\[ \ln y = \frac{1}{x} \ln x \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \). For the left-hand side, we use implicit differentiation:

\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]

For the right-hand side, we will use the product rule since we have \( \frac{1}{x} \ln x \):

\[ \frac{d}{dx}\left(\frac{1}{x} \ln x\right) = \frac{d}{dx}(\ln x) \cdot \frac{1}{x} + \ln x \cdot \frac{d}{dx}\left(\frac{1}{x}\right) \]

Calculating these derivatives:

1. \( \frac{d}{dx}(\ln x) = \frac{1}{x} \)
2. \( \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \)

Putting it all together:

\[ \frac{d}{dx}\left(\frac{1}{x} \ln x\right) = \frac{1}{x} \cdot \frac{1}{x} + \ln x \cdot \left(-\frac{1}{x^2}\right) = \frac{1}{x^2} - \frac{\ln x}{x^2} \]

### Step 4: Set the derivatives equal to each other
Now we can set the derivatives equal:

\[ \frac{1}{y} \frac{dy}{dx} = \frac{1}{x^2} - \frac{\ln x}{x^2} \]

### Step 5: Solve for \( \frac{dy}{dx} \)
To isolate \( \frac{dy}{dx} \), multiply both sides by \( y \):

\[ \frac{dy}{dx} = y \left(\frac{1}{x^2} - \frac{\ln x}{x^2}\right) \]

### Step 6: Substitute back for \( y \)
Recall that \( y = x^{\frac{1}{x}} \):

\[ \frac{dy}{dx} = x^{\frac{1}{x}} \left(\frac{1}{x^2} - \frac{\ln x}{x^2}\right) \]

### Step 7: Factor out common terms
We can factor out \( \frac{1}{x^2} \):

\[ \frac{dy}{dx} = \frac{x^{\frac{1}{x}}}{x^2} (1 - \ln x) \]

### Final Answer
Thus, the derivative of \( y = x^{\frac{1}{x}} \) is:

\[ \frac{dy}{dx} = \frac{x^{\frac{1}{x}} (1 - \ln x)}{x^2} \]

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