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

1.	If `x^(y)=e^(x-y)`, prove that `(dy)/(dx)=(logx)/((1+logx)^(2)).`
Answer» We have `x^(y)=e^(x-y)rArr ylog x=(x-y)` `rArr (1+logx)y=x` `rArry=(x)/((1+logx))." …(i)"` On differentiating both sides of (i) 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))=((1+logx-1))/((1+logx)^(2))=(logx)/((1+logx)^(2)).`

Discussion

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