Differentiate the functions given w.r.t. x:
`(x+1/x)^x+x^((1+1/x))`

Explanation

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

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

1. Take the natural logarithm of both sides:
\[ \ln u = x \ln \left( x + \frac{1}{x} \right) \]
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