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Question
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
Options
`(x(xlogy - y))/(y(ylogx - x)`
`(y(xlogy - y))/(x(ylogx - x)`
`(y^2(1 - logx))/(x^2(1 - logy)`
`(y(1 - logx))/(x(1 - logy)`
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Solution
`(y(xlogy - y))/(x(ylogx - x)`
xy = yx ∴ y log x = x log y
`∴ y xx (1)/x + (logx)"dy"/"dx" = x xx (1)/y"dy"/"dx" + logy`
`∴ (log x - x/y)"dy"/"dx" = logy - y/x`
`∴ ((ylog x - x)/y)"dy"/"dx" = (xlogy - y)/x`
`∴ "dy"/"dx" = (y(xlogy - y))/(x(ylogx - x))`.
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