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Question
If x5· y7 = (x + y)12 then show that, `dy/dx = 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 * dy/dx = 12 * 1/(x + y) * d/dx (x + y)`
∴ `5/x + 7/y * dy/dx = 12/(x + y) [1 + dy/dx]`
∴ `5/x + 7/y * dy/dx = 12/(x + y) + 12/(x + y) * dy/dx`
∴ `[7/y - 12/(x + y)] dy/dx = 12/(x + y) - 5/x`
∴ `[(7x + 7y - 12y)/(y (x + y))] dy/dx = (12x - 5x - 5y)/(x(x + y))`
∴ `[(7x - 5y)/(y(x + y))] dy/dx = [(7x - 5y)/(x(x + y))]`
∴ `dy/dx = [(7x - 5y)/(x(x + y))] xx (y(x + y))/(7x - 5y)`
∴ `dy/dx = y/x`
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