If `x^(y)=e^(x-y)`, prove that `(dy)/(dx)=(logx)/((1+logx)^(2)).`

Explanation

To prove that \(\frac{dy}{dx} = \frac{\log x}{(1 + \log x)^2}\) given the equation \(x^y = e^{x - y}\), we will follow these steps:

### Step 1: Take the natural logarithm of both sides
Starting with the equation:
\[ x^y = e^{x - y} \]
Taking the natural logarithm (log) of both sides, we get:
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