Given y= ⁻¹ x , then d^2 y d x^2 in terms of ' y ' is

Step-by-step solution

Given y = ⁻¹ x , we have x = y . Differentiating both sides with respect to x , we get dy dx = 1 1 + x^2 . Substituting x = y , we obtain dy dx = 1 1 + ^2 y = 1 ^2 y = ^2 y . Now, differentiating dy dx = ^2 y with respect to x using the chain rule: d^2 y dx^2 = d dx ( ^2 y) = d dy ( ^2 y) dy dx . d^2 y dx^2 = (2 y (- y)) ^2 y . d^2 y dx^2 = -2 y y ^2 y . Using the identity 2y = 2 y y , we get d^2 y dx^2 = - 2y ^2 y . Answer: - 2 y ^2 y