Question

If x⁵· y⁷ = (x + y)¹² then show that, `dy/dx = y/x`

Answer 1

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`