Differentiate the following w.r.t. x :
`e^(-x)`

Answer

To differentiate the function \( y = e^{-x} \) with respect to \( x \), we will follow these steps:

### Step 1: Write down the function
Let
\[ y = e^{-x} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( e^{u} \) with respect to \( u \) is \( e^{u} \), and we multiply by the derivative of \( u \) with respect to \( x \).

Here, \( u = -x \), so we have:
\[ \frac{dy}{dx} = \frac{d}{dx}(e^{-x}) = e^{-x} \cdot \frac{d}{dx}(-x) \]

### Step 3: Calculate the derivative of \( -x \)
The derivative of \( -x \) with respect to \( x \) is:
\[ \frac{d}{dx}(-x) = -1 \]

### Step 4: Substitute back into the derivative
Now substituting this back into our derivative expression:
\[ \frac{dy}{dx} = e^{-x} \cdot (-1) = -e^{-x} \]

### Final Answer
Thus, the derivative of \( e^{-x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -e^{-x} \]