3.8: Derivatives of Inverse Functions
- Page ID
- 144212
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- Find \(\arcsin\left(\dfrac{1}{2}\right)\).
- Find \(\arctan(-1)\).
- Find \(\sec^{-1}(-2)\).
- Find \(\cot^{-1}(0)\).
- Find \(\cos^{-1}\left(\cos\left(\dfrac{\pi}{4}\right)\right)\).
- Find \(\arcsin\left(\sin\left(\dfrac{2\pi}{3}\right)\right)\).
- Find \(\sec^{-1}\left(\sec\left(-\dfrac{\pi}{6}\right)\right)\).
- Show that \(\dfrac{d}{dx}\arcsin x = \dfrac{1}{\sqrt{1 - x^2}}\).
- Show that \(\dfrac{d}{dx}\cos^{-1} x = \dfrac{-1}{\sqrt{1 - x^2}}\).
- Show that \(\dfrac{d}{dx}\arctan x = \dfrac{1}{1 + x^2}\).
- Show that \(\dfrac{d}{dx}\cot^{-1} x = \dfrac{-1}{1 + x^2}\).
- Show that \(\dfrac{d}{dx}\text{arcsec} x = \dfrac{1}{|x|\sqrt{x^2 - 1}}\).
- Show that \(\dfrac{d}{dx}\csc^{-1} x = \dfrac{-1}{|x|\sqrt{x^2 - 1}}\).
- Given \(y = \arcsin(x^2)\), find \(\dfrac{dy}{dx}\).
- Given \(y = \cos^{-1}\Bigl(\sin (x^3)\Bigr)\), find \(\dfrac{dy}{dx}\).
- Given \(y = \sec^{-1}\left(\dfrac{1}{x}\right)\), find \(\dfrac{dy}{dx}\).
- Given \(y = \arccos(e^x) \cdot \arcsin(e^x)\), find \(\dfrac{dy}{dx}\).
- Given \(y = \ln \Bigl((\sin^{-1} x)^2\Bigr)\), find \(\dfrac{dy}{dx}\).
- Given \(y = \tan^{-1} \sqrt{9 - x^2}\), find \(\dfrac{dy}{dx}\).