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Question
Find the derivatives of the following:
xy = yx
Solution
xy = yx
Taking log on both sides
log xy = log yx
y log x = x log y
Differentiate with respect to x
`y xx 1/x + (log x) ("d"y)/("d"x) = x * 1/y xx ("d"y)/("d"x) + (log y) 1`
`y/x + (log x) ("d"y)/("d"x) = x/y * ("d"y)/("d"x) + log y`
`(log ) ("d"y)/("d"x) - x/y ("d"y)/("d"x) = log y - y/x`
`(log x - x/y) ("d"y)/("d"x) = log y - y/x`
`(y log x - x)/y * ("d"y)/("d"x) = (x log y - y)/x`
`("d"y)/("d"x) = y/x * (x log y - y)/(y log x - x)`
`("d"y)/("d"x) = (y (x log y - y))/(x(y log x - x))`
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