Differentiate ${cos}^{- 1} (2 x^{2} - 1)$ with respect to ${cos}^{- 1} x$ .

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Solution

We have ${cos}^{- 1} (2 x^{2} - 1) = 2 {cos}^{- 1} x$ .
Let $y = {cos}^{- 1} (2 x^{2} - 1) = 2 {cos}^{- 1} x$ ......(1).
Again let $z = {cos}^{- 1} x = \frac{y}{2}$ . [ Using (1)].....(2).
We are to find $\frac{d y}{d z}$ .
Now from (2) we get, $\frac{d y}{d z} = 2$ .

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