Question

The derivative of sec-1 12x2-1 with respect to 1-x2 at x=12 is:

Answer 1

Given: The function, sec-1 12x2-1
We have to find it's derivative with respect to 1-x2 at x=12
Let y = sec-1 12x2-1= cos-1(2x2 - 1) [Since sec-11θ = cos-1θ]
Substituting x = cos θ ,
we get y = cos-1(2cos2θ - 1)= cos-1(cos2θ ) [Using trigonometric formula of cosine function ]
= 2θ
= 2 cos1
Differentiating w.r.t. x, we get
dydx=-21-x2 [Using standard derivatives-4 ]
Let z=1-x2
Differentiating w.r.t. x, we get
dzdx=−x1−x2∴dydz=dydxdzdx=−21−x2−x1−x2=2x(dydz)at x=12=2(12)=4
Hence, the answer is 4.00