Find (d y)/(d x) if [sec ^-1(x^2-1)]^5 ( mart )

To find for the given function \( \left[\sec^{-1}\left(x^2 - 1\right)\right]^5 \), we can use the chain rule and the derivative of the inverse secant function.Let's define \( u = \sec^{-1}(x^2 - 1) \). Then, .First, we need to find . To do this, we'll differentiate the inner function \( \sec^{-1}(x^2 - 1) \) with respect to .The derivative of \( \sec^{-1}(u) \) with respect to is . Now, let's find by applying the chain rule: Now, let's find by differentiating with respect to : Finally, we can find by applying the chain rule: Substituting back \( u = \sec^{-1}(x^2 - 1) \), we get: Therefore, the derivative of the given function is: