Question

If log (x + y) = log (xy) + 3, then `dy/dx` = ______. 

Options

• `(y/x)^2`

• `-(y/x)^2`

• `(x/y)^2`

• `-(x/y)^2`

Answer 1

If log (x + y) = log (xy) + 3, then `dy/dx` = `underlinebb(-(y/x)^2)`.

Explanation:

We have, log (x + y) = log (xy) + 3

`\implies` log (x + y) = log x + log y + 3   ...(i)

On differentiating both sides of equation (i) w.r.t. x, we get

`1/(x + y) [1 + dy/dx] = 1/x + 1/y dy/dx`

`\implies 1/(x + y) + dy/dx (1/(x + y)) = 1/x + 1/y (dy/dx)`

`\implies dy/dx [1/(x + y) - 1/y] = 1/x - 1/(x + y)`

`\implies dy/dx [(y - x - y)/(y(x + y))] = (x + y - x)/(x(x + y))`

`\implies dy/dx ((-x)/(y(x + y))) = y/(x(x + y))`

`\implies dy/dx = - (y/x)^2`