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"If "x^(y)=e^(x-y)," prove that "(dy)/(d...

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

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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 logarithm of both sides We start with the equation: \[ x^y = e^{x - y} \] Taking the natural logarithm on both sides gives: ...
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