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

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

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`x^(y)=e^(x-y) therefore logx^(y)=loge^(x-y)`
`therefore ylogx=(x-y)loge`
`therefore ylogx=x-y" "...[becauseloge=1]`
`therefore y+ylogx=x thereforey(1+logx)=x`
`thereforey=(x)/(1+logx)`
Differentiating both sides w.r.t. x, we get,
`(dy)/(dx)=((1+logx).(d)/(dx)(x)-x.(d)/(dx)(1+logx))/((1+logx)^(2))`
`=((1+logx)(1)-x((1)/(x)))/((1+logx)^(2))=(logx)/((1+logx)^(2))`
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