If `f (x) =x^(1//x) ,` then f' ' (e ) is equal to

Options

A. ` e^(1//e)`
B. `e^(1//e) -2`
C. ` 2^(1//(e-3)`
D. `-e^((1//e-3))`

Answer

To find \( f''(e) \) for the function \( f(x) = x^{1/x} \), we will follow these steps:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of \( f(x) \):
\[ \ln(f(x)) = \ln(x^{1/x}) \]
Using the logarithmic identity \( \ln(a^b) = b \ln(a) \), we can simplify the right-hand side:
\[ \ln(f(x)) = \frac{1}{x} \ln(x) \]

### Step 2: Differentiate both sides
Now we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\ln(f(x))) = \frac{d}{dx}\left(\frac{1}{x} \ln(x)\right) \]
Using the chain rule on the left side, we have:
\[ \frac{1}{f(x)} f'(x) = \frac{d}{dx}\left(\frac{1}{x} \ln(x)\right) \]
For the right side, we apply the quotient rule:
\[ \frac{d}{dx}\left(\frac{\ln(x)}{x}\right) = \frac{x \cdot \frac{1}{x} - \ln(x) \cdot 1}{x^2} = \frac{1 - \ln(x)}{x^2} \]
Thus, we have:
\[ \frac{1}{f(x)} f'(x) = \frac{1 - \ln(x)}{x^2} \]

### Step 3: Solve for \( f'(x) \)
Multiplying both sides by \( f(x) \):
\[ f'(x) = f(x) \cdot \frac{1 - \ln(x)}{x^2} \]
Substituting \( f(x) = x^{1/x} \):
\[ f'(x) = x^{1/x} \cdot \frac{1 - \ln(x)}{x^2} \]

### Step 4: Differentiate \( f'(x) \) to find \( f''(x) \)
Now we differentiate \( f'(x) \):
\[ f''(x) = \frac{d}{dx}\left(x^{1/x} \cdot \frac{1 - \ln(x)}{x^2}\right) \]
Using the product rule:
\[ f''(x) = \left( \frac{d}{dx}(x^{1/x}) \cdot \frac{1 - \ln(x)}{x^2} \right) + \left( x^{1/x} \cdot \frac{d}{dx}\left(\frac{1 - \ln(x)}{x^2}\right) \right) \]

### Step 5: Calculate \( f''(e) \)
We need to evaluate \( f''(e) \). First, we find \( f(e) \):
\[ f(e) = e^{1/e} \]
Next, we substitute \( x = e \) into our expressions for \( f'(x) \) and \( f''(x) \).

1. Calculate \( f'(e) \):
\[ f'(e) = e^{1/e} \cdot \frac{1 - \ln(e)}{e^2} = e^{1/e} \cdot \frac{1 - 1}{e^2} = 0 \]

2. Now, we need to differentiate \( f'(x) \) again and substitute \( x = e \) to find \( f''(e) \). This involves substituting \( e \) into our derived expression for \( f''(x) \).

After performing the calculations, we find:
\[ f''(e) = -e^{1/e} \cdot \frac{1}{e^4} \]
This simplifies to:
\[ f''(e) = -\frac{e^{1/e}}{e^4} = -e^{1/e - 4} \]

### Final Answer
Thus, the value of \( f''(e) \) is:
\[ f''(e) = -e^{1/e - 4} \]