The derivative of cos - 1 ( 2 x 2 - 1 ) w.r.t. cos - 1 x is:

Question

The derivative of $\cos^{- 1} (2 x^{2} - 1)$ w.r.t. $\cos^{- 1} x$ is:

Smart Achievers · Accepted Answer

(1)

Ans. 2

Explanation:

$Let u = \cos^{- 1} (2 x^{2} - 1) and v = \cos^{- 1} x$

$∴$ $\frac{d u}{d x} = \frac{- 1}{\sqrt{1 - {(2 x^{2} - 1)}^{2}}} \cdot 4 x = \frac{- 4 x}{\sqrt{1 - (4 x^{4} + 1 - 4 x^{2})}}$

$= \frac{- 4 x}{\sqrt{- 4 x^{4} + 4 x^{2}}} = \frac{- 4 x}{\sqrt{4 x^{2} (1 - x^{2})}} = \frac{- 2}{\sqrt{1 - x^{2}}}$

$and$ $\frac{d v}{d x} = \frac{- 1}{\sqrt{1 - x^{2}}}$

$∴$ $\frac{d u}{d v} = \frac{\frac{d u}{d x}}{\frac{d v}{d x}} = \frac{- 2 / \sqrt{1 - x^{2}}}{- 1 / \sqrt{1 - x^{2}}} = 2$

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