Question

find the derivative of sec^-1(1/2x^2-1) wrt sqrt 1-x^2 at x=1/2

Answer 1

Let u = sec−1[12x2−1]put x = cos θSo, u = sec−1[12 cos2θ − 1]⇒u = sec−1[1cos 2θ]⇒u = sec−1(sec 2θ)⇒u = 2θ⇒u = 2 cos−1x⇒dudx = −21−x2√Let v = 11−x2⇒dvdx = 2x(1−x2)2Now,dudv = dudx×dxdv = −21−x2√ × (1−x2)22x = −(1−x2)3/2x[dudv]x=1/2 = −(3/4)3/21/2 = −33√/81/2 = −33√4