If `x^(2) = e^(x-y) `, then `(dy)/(dx)` at x = 1 is ……..

Options

A. e
B. 1
C. 0
D. `-1`

Answer

C

Explanation

Given have, `x^(y) = e^(x - y)`
taking log on both sides, we get
y logx = (x - y) log e = (x - y)
When x = 1, then (log 1) = (1 - y)
`implies y = 1`
On differentiating both sides,
`y((1)/(x)) + log x. (dy)/(dx) = 1 - (dy)/(dx)`
`implies (dy)/(dx) (logx + 1) = 1 - (y)/(x)`
`implies (dy)/(dx) (logx + 1) (x - y)/(x)`
`implies (dy)/(dx) = ((x - y))/(x(logx + 1))`
When x =1, then
`((dy)/(dx)) = (1 -1)/(1(log1 +1)) = 0`