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Question
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
Solution
We have , `xy = e^(x-y)`
Taking log on both sides,
`log (xy) = log (e^(x-y))`
⇒ `log x + log y= (x-y)log e`
⇒ `logx + log y = (x - y) xx 1`
⇒ `log x+ log y = x-y`
⇒ `d/dx (log x) + d/dx (log y ) = d/(dr) (x) - dy/dx`
⇒ `1/x + 1/y dy/dx = 1 -dx /dx`
⇒` ( 1 +1/y) dy/dx = 1 -1/x`
⇒ `(y+1/y)dy/dx = (x-1)/x`
⇒ `dy/dx = (y(x-1))/(x(y+1))`
Hence proved.
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