For the following three (03) items: Let y = f(x) = x ⁻¹ x 1 - x^2 + 1 - x^2 . What is d^2y dx^2 at x = 0 equal to?

Step-by-step solution

Given y = x ⁻¹ x 1 - x^2 + 1 - x^2 We can rewrite the function as: y = x ⁻¹ x 1 - x^2 + 1 2 (1 - x^2) Differentiating with respect to x using the quotient and chain rules: dy dx = ( ⁻¹ x + x 1 - x^2 ) 1 - x^2 - x ⁻¹ x ( -x 1 - x^2 ) 1 - x^2 + 1 2 ( -2x 1 - x^2 ) dy dx = 1 - x^2 ⁻¹ x + x + x^2 ⁻¹ x 1 - x^2 1 - x^2 - x 1 - x^2 dy dx = (1 - x^2) ⁻¹ x + x 1 - x^2 + x^2 ⁻¹ x (1 - x^2)^ 3/2 - x 1 - x^2 dy dx = ⁻¹ x + x 1 - x^2 (1 - x^2)^ 3/2 - x 1 - x^2 dy dx = ⁻¹ x (1 - x^2)^ 3/2 + x 1 - x^2 - x 1 - x^2 = ⁻¹ x (1 - x^2)