For the following three (03) items: Let y = f(x) = x ⁻¹ x 1 - x^2 + 1 - x^2 . What is the slope of the tangent to the curve y = f(x) at x = 0.5 ?

Step-by-step solution

Given y = f(x) = 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 product rule and chain rule: dy dx = d dx ( x 1 - x^2 ) ⁻¹ x + x 1 - x^2 d dx ( ⁻¹ x) + 1 2 1 1 - x^2 (-2x) dy dx = [ 1 1 - x^2 - x ( -x 1 - x^2 ) 1 - x^2 ] ⁻¹ x + x 1 - x^2 1 1 - x^2 - x 1 - x^2 dy dx = [ 1 - x^2 + x^2 (1 - x^2)^ 3/2 ] ⁻¹ x + x 1 - x^2 - x 1 - x^2 dy dx = ⁻¹ x (1 - x^2)^ 3/2 To find the slope of the tangent at x = 0.5 = 1 2 , we substitute this va