3.8: Derivatives of Inverse Functions
- Page ID
- 144212
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 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 \(\dfrac{dy}{dx}\).