3.3: Differentiation Rules
- Page ID
- 143971
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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 \(f(x) = 14\), find \(f'(x)\).
- Given \(g(x) = x^5\), find \(g'(x)\).
- Given \(h(x) = \pi^3\), find \(h'(x)\).
- Given \(y = 2x^3 + 7\), find \(y'\).
- Given \(f(x) = 3x^2 - x + 4\), find \(\dfrac{df}{dx}\).
- Given \(g(x) = \sqrt{2}x^5 - x^{\pi}\), find \(\dfrac{dg}{dx}\).
- Given \(h(x) = \dfrac{1}{2}(x^3 + x^2 + x + 1)\), find \(\dfrac{dh}{dx}\).
- Given \(y = \dfrac{2x^3 - 4}{3}\), find \(\dfrac{dy}{dx}\).
- Given \(f(x) = 2x^{-3} + 4\), find \(f'(x)\).
- Given \(g(x) = \dfrac{2}{x} + \dfrac{x}{2}\), find \(g'(x)\).
- Given \(h(x) = \sqrt{x} - \dfrac{1}{\sqrt{x}}\), find \(h'(x)\).
- Given \(y = \dfrac{1}{2x^{3/2}} + 2\sqrt{x}\), find \(y'(x)\).
- Given \(f(x) = \sqrt[\leftroot{5} \uproot{2} \scriptstyle 3]{\dfrac{8}{x^2}}\), find \(\dfrac{df}{dx}\).
- Given \(g(t) = (t + 1)^2\), find \(\dfrac{dg}{dt}\).
- Given \(h(t) = -5\left(t^3 - 3t^2 - \dfrac{1}{t}\right)\), find \(\dfrac{dh}{dt}\).
- Given \(x = \dfrac{t^2 + 1}{t}\), find \(\dfrac{dx}{dt}\).
- Given \(f(x) = (x^2 + 2x)(2x^3 + 3x^2 - 4)\), find \(f'(x)\).
- Given \(g(x) = (1 + x + x^2)(x^{-1} + x^{-2})\), find \(g'(x)\).
- Given \(h(x) = \left(\dfrac{2}{x^3} - \dfrac{1}{x^5}\right)(x^2 - 5x + 1)\), find \(h'(x)\).
- Given \(y = \dfrac{1}{x^2 + 1}\), find \(y'\).
- Given \(f(x) = \dfrac{x^2 + \sqrt{x}}{x^2 - 1}\), find \(\dfrac{df}{dx}\).
- Given \(g(x) = 3 - \dfrac{2x^{-2} - 4}{x^4}\), find \(\dfrac{dg}{dx}\).
- Given \(h(x) = \dfrac{(x^2 + 1)(\sqrt{x} - 2)}{2x + 3}\), find \(\dfrac{dh}{dx}\).
- Given \(y = (1 - \sqrt{x})\left(\dfrac{x}{x + 1}\right)\), find \(\dfrac{dy}{dx}\).
- Given \(f(x) = |x|\), find \(f'(x)\). [Hint: Consider first the case where \(x > 0\), then separately consider the case where \(x < 0\). For each case, consider how \(|x|\) might be simplified.]
- Find the equation of the line tangent to the graph of \(g(x) = \dfrac{1}{x}\) at \(x = \dfrac{1}{2}\).
- Find the equation of the line tangent to the graph of \(h(x) = \dfrac{x}{1 + x}\) at \(x = 1\).
- Given \(f(x) = \dfrac{1}{6}x^3 - \dfrac{1}{2}x^2 + x - 4\), find \(f'(x)\) and \(f''(x)\).
- Given \(y = \dfrac{x + 1}{x - 1}\), find \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\).