Find the derivative of sec-1(1/2x2-1) w r t (1-x2)-1 at x=1/2 - Maths - Application of Derivatives

Question

Find the derivative of sec-1(1/2x2-1) w r t
(1-x2)-1 at x=1/2

Manbar Singh · Accepted Answer

Let u = sec-112x2-1put x = cos θSo, u = sec-112 cos2θ - 1⇒u = sec-11cos 2θ⇒u = sec-1sec 2θ⇒u = 2θ⇒u = 2 cos-1x⇒dudx = -21-x2Let v = 11-x2⇒dvdx = 2x1-x22Now,dudv = dudx×dxdv = -21-x2 × 1-x222x = -1-x23/2xdudvx=1/2 = -3/43/21/2 = -33/81/2 = -334

Find the derivative of sec-1(1/2x2-1) w.r.t. (1-x2)-1 at x=1/2.

Find the derivative of sec^-1(1/2x²-1) w.r.t. (1-x²)^-1 at x=1/2.