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Question
Find `(dy)/(dx)`, if xy = yx
Solution
Given xy = yx
Taking logarithm of both sides, we get
log xy = log yx
∴ y log x = x log y
Differentiating both sides w.r.t.x, we get
`d/(dx)(ylogx) = d/(dx)(xlogy)`
∴ `y.d/(dx)(logx) + d/(dx)(y) = x.d/(dx)(logy) + logy. d/(dx)(x)`
∴ `y. 1/x + logx.(dy)/(dx) = x. 1/y.(dy)/(dx) + logy.1`
∴ `(logx - x/y)(dy)/(dx) = (logy - y/x)`
∴ `((ylogx - x)/y) (dy)/(dx) = (xlogy - y)/x`
∴ `(dy)/(dx) = ((xlogy - y)/x) xx (y/(ylogx - x))`
∴ `(dy)/(dx) = y/x((xlogy - y)/(ylogx - x))`
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