If the derivative of ⁻¹(a + bx) with respect to x at x = 0 is 1 , then a^6 - b^3 =

Step-by-step solution

Let y = ⁻¹(a + bx) Differentiating with respect to x , we get: dy dx = 1 1 + (a + bx)^2 b Given that at x = 0 , dy dx = 1 : b 1 + a^2 = 1 b = 1 + a^2 a^2 - b = -1 Cubing both sides, we get: (a^2 - b)^3 = (-1)^3 a^6 - b^3 - 3a^2b(a^2 - b) = -1 Substituting a^2 - b = -1 : a^6 - b^3 - 3a^2b(-1) = -1 a^6 - b^3 + 3a^2b = -1 a^6 - b^3 = -1 - 3a^2b Answer: -1 - 3a^2 b