NDA Mathematics Question (2025) — Solution

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 value into the derivative: dy dx |_ x=1/2 = ^ -1 (1/2) (1 - (1/2)^2)^ 3/2 dy dx |_ x=1/2 = /6 (3/4)^ 3/2 = /6 3 3 /8 dy dx |_ x=1/2 = 8 18 3 = 4 9 3 Rationalizing the denominator by multiplying the numerator and denominator by 3 : dy dx |_ x=1/2 = 4 3 27