If xy = yx, find dy/dx

1.	If xy = yx, find dy/dx
Answer» Given, x^y = y^x Taking logarithm on both sides, we get y log x = x log y Differentiating both sides, w.r.t. x y.(1/x) + log x(dy/dx) = x.(1/y).(dy/dx) + log y. 1 (y/x) + (log x).(dy/dx) = (x/y).(dy/dx) + log y (dy/dx)[log x - (x/y)] = log y - (y/x) (dy/dx)[(y log x - x)/y] = (x log y - y)/x ∴ dy/dx = y(x log y - y)/x(y log x - x)

Answer»

Given, x^y = y^x

Taking logarithm on both sides, we get

y log x = x log y

Differentiating both sides, w.r.t. x

y.(1/x) + log x(dy/dx) = x.(1/y).(dy/dx) + log y. 1

(y/x) + (log x).(dy/dx) = (x/y).(dy/dx) + log y

(dy/dx)[log x - (x/y)] = log y - (y/x)

(dy/dx)[(y log x - x)/y] = (x log y - y)/x

∴ dy/dx = y(x log y - y)/x(y log x - x)

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