`(e^(x))/((1+x))`

Answer

To differentiate the function \( y = \frac{e^x}{1+x} \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then its derivative is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = e^x \) and \( v = 1 + x \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = e^x \)
- \( v = 1 + x \)

### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find the derivatives of \( u \) and \( v \):
- \( \frac{du}{dx} = e^x \) (the derivative of \( e^x \) is \( e^x \))
- \( \frac{dv}{dx} = 1 \) (the derivative of \( 1 + x \) is \( 1 \))

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

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

### Step 4: Simplify the Expression
Now, we simplify the numerator:

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

This simplifies to:

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

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

\[ \frac{dy}{dx} = \frac{x e^x}{(1+x)^2} \]

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