Question

If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.

विकल्प

• `x/y`

• `-y/x`

• `-x/y`

• `y/x`

Answer 1

If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = `underlinebb(y/x)`.

Explanation:

Since, `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2

∴ log (x³ – y³) – log (x³ + y³) = 2

`\implies` log (x³ – y³) = 2 + log (x³ + y³)

Differentiating both sides w.r.t. x

`\implies 1/(x^3 - y^3)[3x^2 - 3y^2 dy/dx] = 1/(x^3 + y^3)[3x^2 + 3y^2 dy/dx]`

`\implies (3x^2)/(x^3 - y^3) - (3y^2)/(x^3 - y^3) dy/dx = (3x^2)/(x^3 + y^3) + (3y^2)/(x^3 + y^3) dy/dx`

`\implies (3x^2)/(x^3 - y^3) - (3x^2)/(x^3 + y^3) = [(3y^2)/(x^3 + y^3) + (3y^2)/(x^3 - y^3)] dy/dx`

`\implies 3x^2[1/(x^3 - y^3) - 1/(x^3 + y^3)] = 3y^2[1/(x^3 + y^3) + 1/(x^3 - y^3)]dy/dx`

`\implies 3x^2[(2y^3)/((x^3 - y^3)(x^3 + y^3))] = 3y^2[(2x^3)/((x^3 + y^3)(x^3 - y^3))]dy/dx`

`implies y/x = dy/dx`