[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

With the aid of the rules for differentiation given in this section, compute [math]f^\prime = \ddxof f[/math] for each of the following functions.

• [a [math]f(x) = 3x^2 + 4x + 1[/math]
• [math]f(x) = x^2 (x + 1)[/math]
• [math]f(x) = x^3 (x + 2)^2[/math]
• [math]f(x) = (x^2 - 4)(x^2 + 2x + 3)[/math]
• [math]f(x) = 2x^2 + \frac1{3x^3}[/math]
• [math]f(x) = \frac{2x}{2-x}[/math]
• [math]f(x) = \frac{2x}{(2-x)^2}[/math]
• [math]f(x) = \frac{x^3}{x^5 + 1}[/math]
• [math]f(x) = \left( \frac{3-x}{3+x} \right)^2[/math]
• [math]f(x) = (x^2 + 1)^3[/math]
• [math]f(x) = \frac{2x+1}{x^2+x}[/math]
• [math]f(x) = (x^2 + 1)^{-1}[/math]
• [math]f(x) = (x + x^{-1})^2[/math]
• [math]f(x) = (x-a)(x-b)(x-c)[/math]