The second order derivative of $\sec^{-1}\left(\frac{1}{2x^2 - 1}\right)$ with respect to $\cos^{-1}(2x^2 - 1)$, where $0<x<\frac{1}{\sqrt{2}}$ is
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Before jumping into messy chain rule calculations for "derivative of $f(x)$ with respect to $g(x)$", always check if $f(x)$ can be simplified or rewritten directly in terms of $g(x)$ using algebraic or trigonometric identities. Recognizing $u=v$ reduces a 5-minute problem to a 5-second one.