Skip to main content\(\require{cancel}\renewcommand{\neg}{ {\sim} }
\newcommand{\limp}{ {\;\Rightarrow\;} }
\newcommand{\nimp}{ {\;\not\Rightarrow\;} }
\newcommand{\liff}{ {\;\Leftrightarrow\;} }
\newcommand{\niff}{ {\;\not\Leftrightarrow\;} }
\newcommand{\st}{ {\mbox{ s.t. }} }
\newcommand{\es}{ {\varnothing}}
\newcommand{\ds}{ \displaystyle }
\newcommand{\pow}[1]{ \mathcal{P}\left(#1\right) }
\newcommand{\set}[1]{ \left\{#1\right\} }
\newcommand{\bdiff}[2]{ \frac{\mathrm{d}}{\mathrm{d}#2} \left( #1 \right)}
\newcommand{\ddiff}[3]{ \frac{\mathrm{d}^#1#2}{\mathrm{d}{#3}^#1}}
\newcommand{\pdiff}[2]{ \frac{\partial #1}{\partial #2}}
\newcommand{\bbbn}{\mathbb{N}}
\newcommand{\bbbr}{\mathbb{R}}
\newcommand{\bbbp}{\mathbb{P}}
\newcommand{\De}{\Delta}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\veps}{\varepsilon}
\newcommand{\half}{\tfrac{1}{2}}
\newcommand{\ts}{ \textstyle }
\newcommand{\diff}[2]{ \frac{\mathrm{d} #1 }{\mathrm{d}#2}}
\newcommand{\difftwo}[2]{ \frac{\mathrm{d^2} #1 }{\mathrm{d}{#2}^2}}
\newcommand{\dee}[1]{\mathrm{d}#1}
\newcommand{\soln}{ \medskip\noindent\emph{Solution.}\ \ \ }
\newcommand{\arccsc}{\mathop{\mathrm{arccsc}}}
\newcommand{\arcsec}{\mathop{\mathrm{arcsec}}}
\newcommand{\arccot}{\mathop{\mathrm{arccot}}}
\newcommand{\erf}{\mathop{\mathrm{erf}}}
\newcommand{\smsum}{\mathop{{\ts \sum}}}
\newcommand{\atp}[2]{ \genfrac{}{}{0in}{}{#1}{#2} }
\newcommand{\vr}{\mathbf{r}}
\newcommand{\vR}{\mathbf{R}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\va}{\mathbf{a}}
\newcommand{\vb}{\mathbf{b}}
\newcommand{\vc}{\mathbf{c}}
\newcommand{\vd}{\mathbf{d}}
\newcommand{\ve}{\mathbf{e}}
\newcommand{\vf}{\mathbf{f}}
\newcommand{\vC}{\mathbf{C}}
\newcommand{\vp}{\mathbf{p}}
\newcommand{\vn}{\mathbf{n}}
\newcommand{\vt}{\mathbf{t}}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vw}{\mathbf{w}}
\newcommand{\vV}{\mathbf{V}}
\newcommand{\vx}{\mathbf{x}}
\newcommand{\vF}{\mathbf{F}}
\newcommand{\vG}{\mathbf{G}}
\newcommand{\vH}{\mathbf{H}}
\newcommand{\vT}{\mathbf{T}}
\newcommand{\vN}{\mathbf{N}}
\newcommand{\vL}{\mathbf{L}}
\newcommand{\vA}{\mathbf{A}}
\newcommand{\vB}{\mathbf{B}}
\newcommand{\vD}{\mathbf{D}}
\newcommand{\vE}{\mathbf{E}}
\newcommand{\vJ}{\mathbf{J}}
\newcommand{\vZero}{\mathbf{0}}
\newcommand{\vPhi}{\mathbf{\Phi}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\Ha}{\hat{\mathbf{a}}}
\newcommand{\he}{\hat{\mathbf{e}}}
\newcommand{\hi}{\hat{\pmb{\imath}}}
\newcommand{\hj}{\hat{\pmb{\jmath}}}
\newcommand{\hk}{\hat{\mathbf{k}}}
\newcommand{\hn}{\hat{\mathbf{n}}}
\newcommand{\hr}{\hat{\mathbf{r}}}
\newcommand{\hvt}{\hat{\mathbf{t}}}
\newcommand{\hN}{\hat{\mathbf{N}}}
\newcommand{\vth}{{\pmb{\theta}}}
\newcommand{\vTh}{{\pmb{\Theta}}}
\newcommand{\vnabla}{ { \mathchoice{\pmb{\nabla}}
{\pmb{\nabla}}
{\pmb{\scriptstyle\nabla}}
{\pmb{\scriptscriptstyle\nabla}} } }
\newcommand{\ha}[1]{\mathbf{\hat e}^{(#1)}}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\vOm}{\pmb{\Omega}}
\newcommand{\vom}{\pmb{\omega}}
\newcommand{\svOm}{\pmb{\scriptsize\Omega}}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\ka}{\kappa}
\newcommand{\la}{\lambda}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\bbbone}{\mathbb{1}}
\DeclareMathOperator{\sgn}{sgn}
\def\tr{\mathop{\rm tr}}
\newcommand{\Atop}[2]{\genfrac{}{}{0pt}{}{#1}{#2}}
\newcommand{\dblInt}{\iint}
\newcommand{\tripInt}{\iiint}
\newcommand{\Set}[2]{\big\{ \ #1\ \big|\ #2\ \big\}}
\newcommand{\rhof}{{\rho_{\!{\scriptscriptstyle f}}}}
\newcommand{\rhob}{{\rho_{{\scriptscriptstyle b}}}}
\newcommand{\hT}{\hat{\mathbf{T}}}
\newcommand{\hB}{\hat{\mathbf{B}}}
\newcommand{\YEaxis}[2]{\draw[help lines] (-#1,0)--(#1,0) node[right]{$x$};
\draw[help lines] (0,-#2)--(0,#2) node[above]{$y$};}
\newcommand{\YEaaxis}[4]{\draw[help lines] (-#1,0)--(#2,0)
node[right]{$x$};
\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};}
\newcommand{\YEtaaxis}[4]{\draw[help lines] (-#1,0)--(#2,0)
node[right]{$t$};
\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};}
\newcommand{\YEtaxis}[4]{\draw[help lines] (-#1,0)--(#2,0)
node[right]{$t$};
\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};}
\newcommand{\YExcoord}[2]{
\draw (#1,.2)--(#1,-.2) node[below]{$#2$};}
\newcommand{\YEycoord}[2]{
\draw (.2,#1)--(-.2,#1) node[left]{$#2$};}
\newcommand{\YEnxcoord}[2]{
\draw (#1,-.2)--(#1,.2) node[above]{$#2$};}
\newcommand{\YEnycoord}[2]{
\draw (-.2,#1)--(.2,#1) node[right]{$#2$};}
\newcommand{\YEstickfig}[3]{
\draw (#1,#2) arc(-90:270:2mm);
\draw (#1,#2)--(#1,#2-.5) (#1-.25,#2-.75)--(#1,#2-.5)--(#1+.25,#2-.75) (#1-.2,#2-.2)--(#1+.2,#2-.2);}
\newcommand{\trigtri}[4]{
\begin{tikzpicture}
\draw (-.5,0)--(2,0)--(2,1.5)--cycle;
\draw (1.8,0) |- (2,.2);
\draw[double] (0,0) arc(0:30:.5cm);
\draw (0,.2) node[right]{$#1$};
\draw (1,-.5) node{$#2$};
\draw (2,.75) node[right]{$#3$};
\draw (.6,1.1) node[rotate=30]{$#4$};
\end{tikzpicture}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section A.3 Table of Derivatives
Throughout this table, \(a\) and \(b\) are constants, independent of \(x\text{.}\)
| \(F(x)\) |
\(F'(x)=\diff{F}{x}\) |
| \(af(x)+bg(x)\) |
\(af'(x)+bg'(x)\) |
| \(f(x)+g(x)\) |
\(f'(x)+g'(x)\) |
| \(f(x)-g(x)\) |
\(f'(x)-g'(x)\) |
| \(af(x)\) |
\(af'(x)\) |
| \(f(x)g(x)\) |
\(f'(x)g(x)+f(x)g'(x)\) |
| \(f(x)g(x)h(x)\) |
\(f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)\) |
| \(\frac{f(x)}{g(x)}\) |
\(\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\) |
| \(\frac{1}{g(x)}\) |
\(-\frac{g'(x)}{g(x)^2}\) |
| \(f\big(g(x)\big)\) |
\(f'\big(g(x)\big)g'(x)\) |
| \(F(x)\) |
\(F'(x)=\diff{F}{x}\) |
| \(a\) |
\(0\) |
| \(x^a\) |
\(ax^{a-1}\) |
| \(g(x)^a\) |
\(ag(x)^{a-1}g'(x)\) |
| \(\sin x\) |
\(\cos x\) |
| \(\sin g(x)\) |
\(g'(x)\cos g(x)\) |
| \(\cos x\) |
\(-\sin x\) |
| \(\cos g(x)\) |
\(-g'(x)\sin g(x)\) |
| \(\tan x\) |
\(\sec^2 x\) |
| \(\csc x\) |
\(-\csc x\cot x\) |
| \(\sec x\) |
\(\sec x\tan x\) |
| \(\cot x\) |
\(-\csc^2 x\) |
| \(e^x\) |
\(e^x\) |
| \(e^{g(x)}\) |
\(g'(x)e^{g(x)}\) |
| \(a^x\) |
\((\ln a)\ a^x\) |
| \(F(x)\) |
\(F'(x)=\diff{F}{x}\) |
| \(\ln x\) |
\(\frac{1}{x}\) |
| \(\ln g(x)\) |
\(\frac{g'(x)}{g(x)}\) |
| \(\log_a x\) |
\(\frac{1}{x\ln a}\) |
| \(\arcsin x\) |
\(\frac{1}{\sqrt{1-x^2}}\) |
| \(\arcsin g(x)\) |
\(\frac{g'(x)}{\sqrt{1-g(x)^2}}\) |
| \(\arccos x\) |
\(-\frac{1}{\sqrt{1-x^2}}\) |
| \(\arctan x\) |
\(\frac{1}{1+x^2}\) |
| \(\arctan g(x)\) |
\(\frac{g'(x)}{1+g(x)^2}\) |
| \(\arccsc x\) |
\(-\frac{1}{|x|\sqrt{x^2-1}}\) |
| \(\arcsec x\) |
\(\frac{1}{|x|\sqrt{x^2-1}}\) |
| \(\arccot x\) |
\(-\frac{1}{1+x^2}\) |
