Differentiate the following: `x+y=ln⁡(e^x-e^y)`

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The variables x and y are related as `x+y= ln⁡((e^x-e^y))`

`x+y= ln⁡((e^x-e^y))`

=> `e^(x + y) = (e^x-e^y)`

=> `e^x*e^y = e^x - e^y`

Using implicit differentiation gives:

`e^x*e^y + e^x*e^y*(dy/dx) = e^x - e^y*(dy/dx)`

=> `(dy/dx)(e^x*e^y + e^y) = (e^x - e^x*e^y)`

=> `dy/dx = (e^x - e^x*e^y)/(e^x*e^y + e^y)`

The required derivative `dy/dx = (e^x - e^x*e^y)/(e^x*e^y + e^y)`

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