The derivative of cos -1(2 x2-1) with respect to cos -1 x is

Question

The derivative of cos -1(2 x2-1) with respect to cos -1 x is (A) 2 (B) (-1/2 √1-x2) (C) (2/x) (D) 1-x2

Tardigrade Admin · Accepted Answer

Let u= cos -1(2 x2-1) (d u/d x)=(-1/√1-(2 x2-1)2) 4 x=(-4 x/√4 x2(1-x2)) =(-4 x/2 x √1-x2)=(-2/√1-x2) v= cos -1 x (d v/d x)=-1 / √1-x2 (d u/d v)=(-2 / √1-x2/-1 / √1-x2)=2

Q. The derivative of $cos^{- 1} (2 x^{2} - 1)$ with respect to $cos^{- 1} x$ is

$2$

$\frac{- 1}{2 1 - x ^{2}}$

$\frac{2}{x}$

$1 - x^{2}$

Q. The derivative of cos−1(2x2−1) with respect to cos−1x is

2

21−x2​−1​

x2​

1−x2

Let u=cos−1(2x2−1) dxdu​=1−(2x2−1)2​−1​4x=4x2(1−x2)​−4x​ =2x1−x2​−4x​=1−x2​−2​ v=cos−1x dxdv​=−1/1−x2​ dvdu​=−1/1−x2​−2/1−x2​​=2

Q. The derivative of $cos^{- 1} (2 x^{2} - 1)$ with respect to $cos^{- 1} x$ is

$2$

$\frac{- 1}{2 1 - x ^{2}}$

$\frac{2}{x}$

$1 - x^{2}$