If `y=(1+1/x)^x` , then `(dy)/(dx)=` (a)`(1+1/x)^x{log(1+1/x)-1/(x+1)}` (b) `(1+1/x)^xlog(1+1/x)` (c) `(x+1/x)^x{log(x+1)-x/(x+1)}` (d) `(x+1/x)^x{log(1+1/x)+1/(x+1)}`

Explanation

To find the derivative of the function \( y = \left(1 + \frac{1}{x}\right)^x \), we will use logarithmic differentiation. Here are the steps:

### Step 1: Take the natural logarithm of both sides
\[ \ln y = \ln\left(\left(1 + \frac{1}{x}\right)^x\right) \]
Using the property of logarithms, we can bring down the exponent:
\[
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