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Question
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
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Solution
x7 . y5 = (x + y)12
Taking log on both sides, we get
log(x7y5) = log(x + y)12
∴ 7log x + 5log y = 12log(x + y)
Differentiating w. r. t. x, we get
`7/x + 5/y* ("d"y)/("d"x) = 12/(x + y)*"d"/("d"x)(x + y)`
∴ `7/x + 5/y*("d"y)/("d"x) = 12/(x + y)(1 + ("d"y)/("d"x))`
∴ `("d"y)/("d"x)(5/y - 12/(x + y)) = 12/(x + y) - 7/x`
∴ `("d"y)/("d"x)((5x - 7y)/(y(x + y))) = (5x - 7y)/(x(x + y))`
∴ `("d"y)/("d"x) = y/x`
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