3.5: The Chain Rule
- Page ID
- 144104
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Given \(f(x) = (x + 1)^{100}\), find \(f'(x)\).
- Given \(y = (7x^2 - 12x + 3)^5\), find \(y'\).
- Given \(g(x) = \sin x^2\), find \(g'(x)\).
- Given \(h(x) = \sin^2 x\), find \(h'(x)\).
- Given \(F(x) = \sqrt{1 - x^2}\), find \(\dfrac{dF}{dx}\).
- Given \(y = \sqrt{3x^2 + 2x - 4}\), find \(y'\).
- Given \(f(x) = \sec(\tan x)\), find \(\dfrac{df}{dx}\).
- Given \(g(x) = \left(2x^3 - \dfrac{1}{x}\right)^{-3}\), find \(\dfrac{dg}{dx}\).
- Given \(h(x) = \dfrac{1}{(5x - 3x^5)^4}\), find \(\dfrac{dh}{dx}\).
- Given \(y = \sqrt{1 + \sqrt{x}}\), find \(\dfrac{dy}{dx}\).
- Given \(f(x) = \cos \sqrt{x}\), find \(f'(x)\).
- Given \(g(x) = \sin(\cos 2x)\), find \(g'(x)\).
- Given \(h(x) = \Bigl(\csc(3x - 2) + \cot(3x - 2)\Bigr)^5\), find \(h'(x)\).
- Given \(y = \tan^2 \left(\dfrac{1}{x}\right)\), find \(y'\).
- Given \(f(x) = \dfrac{1}{\sqrt{\sec x}}\), find \(\dfrac{df}{dx}\).
- Given \(g(x) = \sin(x^2\csc x)\), find \(\dfrac{dg}{dx}\).
- Given \(h(x) = \cos\left(\dfrac{x}{1 + x}\right)\), find \(\dfrac{dh}{dx}\).
- Given \(y = \tan(x^2 + x)\cot(2x - 3)\), find \(\dfrac{dy}{dx}\).
- Given \(f(x) = \dfrac{\sin (2x^3)}{1 - \cos(3x + 4)}\), find \(f'(x)\).
- Given \(g(x) = \sqrt{2x^3 - \cos^2(4x)}\), find \(g'(x)\).
- Given \(h(x) = \sec^5 \left(\sqrt{5x^2 - 4}\right)\), find \(h'(x)\).
- Given \(y(x) = \sin(x^2)\), find \(y''(x)\).
- Given \(f(x) = \tan^2 x\), find \(\dfrac{d^2f}{dx^2}\).
- Find the equation of the line tangent to the graph of \(y = \sqrt{2x^3 - 4x}\) at \(x = 2\).