Maharashtra State BoardHSC Commerce 12th Board Exam
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If x5.y7 = (x + y)12, then show that dydx=yx - Mathematics and Statistics

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Sum

If x5.y7 = (x + y)12, then show that `("d"y)/("d"x) = y/x`

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Solution

x5.y7 = (x + y)12 

Taking logarithm of both sides, we get

log(x5.y7) = log(x + y)12 

∴ log x5 + log y7 = 12 log (x + y)

∴ 5 log x + 7 log y = 12 log (x + y)

Differentiating both sides w.r.t. x, we get

`5*1/x + 7*1/y*("d"y)/("d"x) = 12*1/(x + y)*"d"/("d"x)(x + y)`

∴ `5/x + 7/y*("d"y)/("d"x) = 12/(x + y)(1 + ("d"y)/("d"x))`

∴ `5/x + 7/y*("d"y)/("d"x) = 12/(x + y) + 12/(x + y)*("d"y)/("d"x)`

∴ `7/y*("d"y)/("d"x) - 12/(x + y)*("d"y)/("d"x) = 12/(x + y) - 5/x`

∴ `(7/y - 12/(x + y)) ("d"y)/("d"x) = 12/(x + y) - 5/x`

∴ `[(7x + 7y - 12y)/(y(x + y))]("d"y)/("d"x) = (12x - 5x - 5y)/(x(x + y))`

∴ `[(7x - 5y)/(y(x + y))] ("d"y)/("d"x) = [(7x - 5y)/(x(x + y))]`

∴ `("d"y)/("d"x) = (7x - 5y)/(x(x + y)) xx (y(x + y))/(7x - 5y)`

∴ `("d"y)/("d"x) = y/x`

Concept: The Concept of Derivative - Derivatives of Logarithmic Functions
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