If `y = e^(1//x) " then " (dy)/(dx) =` ?

Options

A. `(1)/(x).e^((1//x-1))`
B. `(-e^(1//x))/(x^(2))`
C. `e^(1//x) log x`
D. none of these

Explanation

To find the derivative of the function \( y = e^{\frac{1}{x}} \), we can use the chain rule for differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = e^{\frac{1}{x}} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we will apply the chain rule. The derivative of \( e^u \) with respect to \( x \) is \( e^u \cdot \frac{du}{dx} \), where \( u = \frac{1}{x} \).

### Step 3: Differentiate \( u = \frac{1}{x} \)
Now, we need to find \( \frac{du}{dx} \):
\[ u = \frac{1}{x} \implies \frac{du}{dx} = -\frac{1}{x^2} \]

### Step 4: Apply the chain rule
Using the chain rule:
\[ \frac{dy}{dx} = e^{\frac{1}{x}} \cdot \frac{du}{dx} = e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2}\right) \]

### Step 5: Simplify the expression
Thus, we have:
\[ \frac{dy}{dx} = -\frac{e^{\frac{1}{x}}}{x^2} \]

### Final Answer
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{e^{\frac{1}{x}}}{x^2} \]

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