`y=(1+x)^(x)`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = (1 + x)^x \), we can follow these steps:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of both sides to simplify the expression:
\[ \ln y = \ln((1 + x)^x) \]

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

### Step 3: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(x \ln(1 + x)) \]

Using the chain rule on the left side and the product rule on the right side:
\[ \frac{1}{y} \frac{dy}{dx} = \ln(1 + x) + x \cdot \frac{1}{1 + x} \]

### Step 4: Simplify the right side
The right side simplifies to:
\[ \frac{1}{y} \frac{dy}{dx} = \ln(1 + x) + \frac{x}{1 + x} \]

### Step 5: Multiply both sides by \( y \)
To isolate \( \frac{dy}{dx} \), multiply both sides by \( y \):
\[ \frac{dy}{dx} = y \left( \ln(1 + x) + \frac{x}{1 + x} \right) \]

### Step 6: Substitute back for \( y \)
Since we know that \( y = (1 + x)^x \), we substitute this back into the equation:
\[ \frac{dy}{dx} = (1 + x)^x \left( \ln(1 + x) + \frac{x}{1 + x} \right) \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = (1 + x)^x \left( \ln(1 + x) + \frac{x}{1 + x} \right) \]

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