University Calculus: Early Transcendentals (3rd Edition)

${\frac{d{csc^{-1}{u}}}{du}}$ = -${\frac{1}{|u|{\sqrt{({(u^2-1)})}}}}$
${csc^{-1}{u}}$ value determined by the given equation: ${csc^{-1}{u}}$ = ${\frac{\pi}{2}}-{sec^{-1}{u}}$ on differentiating both sides: ${\frac{d{csc^{-1}{u}}}{du}}={\frac{d\pi}{2du}}-{\frac{d{sec^{-1}{u}}}{du}}$ differentiation of constant is zero: and ${\frac{d{\sec^{-1}{u}}}{du}={\frac{1}{|u|{\sqrt{({(u^2-1)})}}}}}$ so ${\frac{d{csc^{-1}{u}}}{du}}$ = -${\frac{1}{|u|{\sqrt{({(u^2-1)})}}}}$ thus the final answer is: ${\frac{d{csc^{-1}{u}}}{du}}$ = -${\frac{1}{|u|{\sqrt{({(u^2-1)})}}}}$