Differentiate `(x+1/x)^x + x^(1+1/x)` with respect to `x` .

Explanation

To differentiate the function \( y = \left( x + \frac{1}{x} \right)^x + x^{1 + \frac{1}{x}} \) with respect to \( x \), we will break it down into two parts: \( u = \left( x + \frac{1}{x} \right)^x \) and \( v = x^{1 + \frac{1}{x}} \). Then we will find the derivatives \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) separately and combine them to find \( \frac{dy}{dx} \).

### Step 1: Differentiate \( u = \left( x + \frac{1}{x} \right)^x \)

To differentiate \( u \), we will use logarithmic differentiation.

1. Take the natural logarithm of both sides:
\[
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