If ` y = x^(1/x)` , the value of ` (dy)/(dx)` at x =e is equal to

Options

A. 1
B. 0
C. `-1`
D. `e^((1/e)-2)`

Answer

To find the value of \(\frac{dy}{dx}\) at \(x = e\) for the function \(y = x^{\frac{1}{x}}\), we will follow these steps:

### Step 1: Take the natural logarithm of both sides
We start with the equation:
\[ y = x^{\frac{1}{x}} \]
Taking the natural logarithm on both sides gives:
\[ \ln y = \ln\left(x^{\frac{1}{x}}\right) \]

### Step 2: Use the properties of logarithms
Using the property of logarithms \(\ln(a^b) = b \ln a\), we can rewrite the equation:
\[ \ln y = \frac{1}{x} \ln x \]

### Step 3: Differentiate both sides with respect to \(x\)
Now we differentiate both sides. For the left side, we use the chain rule:
\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]
For the right side, we apply the quotient rule to \(\frac{\ln x}{x}\):
\[ \frac{d}{dx}\left(\frac{\ln x}{x}\right) = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \]

Putting it all together, we have:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{1 - \ln x}{x^2} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Now we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = y \cdot \frac{1 - \ln x}{x^2} \]

### Step 5: Substitute \(y\) back into the equation
Recall that \(y = x^{\frac{1}{x}}\), thus:
\[ \frac{dy}{dx} = x^{\frac{1}{x}} \cdot \frac{1 - \ln x}{x^2} \]

### Step 6: Evaluate at \(x = e\)
Now we substitute \(x = e\):
\[ \frac{dy}{dx} \bigg|_{x=e} = e^{\frac{1}{e}} \cdot \frac{1 - \ln e}{e^2} \]
Since \(\ln e = 1\), we have:
\[ \frac{dy}{dx} \bigg|_{x=e} = e^{\frac{1}{e}} \cdot \frac{1 - 1}{e^2} = e^{\frac{1}{e}} \cdot 0 = 0 \]

Thus, the value of \(\frac{dy}{dx}\) at \(x = e\) is:
\[ \boxed{0} \]