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% Document Info
\author{CROSSANT (CROSSANT)}
\pdfinfo{
  /Title (calculus-derivatives-and-differentiation.pdf)
  /Creator (Cheatography)
  /Author (CROSSANT (CROSSANT))
  /Subject (Calculus Derivatives and Differentiation Cheat Sheet)
}

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        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Calculus Derivatives and Differentiation Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{CROSSANT (CROSSANT)} via \textcolor{DarkBackground}{\uline{cheatography.com/186482/cs/43207/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}CROSSANT (CROSSANT) \\
  \uline{cheatography.com/crossant} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 26th April, 2024.\\
   Updated 4th July, 2024.\\
   Page {\thepage} of \pageref{LastPage}.
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  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{2}

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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Notation}}  \tn
% Row 0
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\mymulticolumn{1}{x{8.4cm}}{Name} \tn 
% Row Count 1 (+ 1)
% Row 1
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% Row 2
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% Row 3
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% Row 4
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% Row Count 5 (+ 1)
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\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{n ∈ ℕ\textasciitilde{}1\textasciitilde{} = \{1,2,3,4,5,...\}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.872 cm} x{1.584 cm} x{1.872 cm} x{1.872 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{8.4cm}}{\bf\textcolor{white}{Derivative Rules}}  \tn
% Row 0
\SetRowColor{LightBackground}
\seqsplit{Formal/Limit} \seqsplit{Definition} of a \seqsplit{Derivative} & lim \textasciitilde{}h-\textgreater{}0\textasciitilde{} \seqsplit{(f(x+h)-f(x))/h} & lim \textasciitilde{}x-\textgreater{}a\textasciitilde{} \seqsplit{(f(x)-f(a))/(x-a)} &  \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
{\bf{Linearity 1: \seqsplit{Constant-Multiple} Rule}} & {\bf{d/dx (kf(x))}} & {\bf{k*d/dx (f)}} & {\bf{kf'}} \tn 
% Row Count 8 (+ 4)
% Row 2
\SetRowColor{LightBackground}
{\bf{Linearity 2: \seqsplit{Sum-Difference} Rule}} & {\bf{d/dx (f(x)±g(x))}} & {\bf{d/dx (f) ± d/dx (g)}} & {\bf{f'±g'}} \tn 
% Row Count 12 (+ 4)
% Row 3
\SetRowColor{white}
{\bf{Product Rule}} & {\bf{d/dx (f(x)*g(x))}} & {\bf{f'g+fg'}} &  \tn 
% Row Count 15 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\seqsplit{Multi-Product} Rule & d/dx \seqsplit{(p(x)*q(x)*r(x)*s(x)*}...) & p'qrs... + pq'rs... + pqr's... + pqrs'... + ... & \seqsplit{pqrs...*(p'/p} + q'/q + r'/r + s'/s + ...) \tn 
% Row Count 20 (+ 5)
% Row 5
\SetRowColor{white}
Quotient Rule & d/dx \seqsplit{(f(x)/g(x))} & (f'g-fg')/g\textasciicircum{}2\textasciicircum{} & g(x)≠0, quotients can be rewritten into products with \seqsplit{sign-flipped} exponents \tn 
% Row Count 28 (+ 8)
% Row 6
\SetRowColor{LightBackground}
{\bf{Chain Rule}} & {\bf{d/dx (f(g(x)))}} & {\bf{f'(g)g'}} &  \tn 
% Row Count 31 (+ 3)
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\par\addvspace{1.3em}

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\mymulticolumn{4}{x{8.4cm}}{\bf\textcolor{white}{Derivative Rules (cont)}}  \tn
% Row 7
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{\bf{Multi-Chain Rule}} & {\bf{d/dx \seqsplit{(p(q(r(s(}...)))))}} & {\bf{p'(q(r(s(...))))*q'(r(s(...)))*r'(s(...))*s'(...)*...}} &  \tn 
% Row Count 7 (+ 7)
% Row 8
\SetRowColor{white}
{\bf{First \seqsplit{Fundamental} Theorem of Calculus (FTC I)}} & {\bf{d/dx (∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}x\textasciicircum{} f(t)dt)}} & {\bf{f(x)}} & {\bf{Derivatives and integrals are inverses of each other}} \tn 
% Row Count 13 (+ 6)
% Row 9
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FTC I Chain Rule 1 & d/dx (∫\textasciitilde{}a\textasciitilde{}\textasciicircum{}v(x)\textasciicircum{}  f(t)dt) & f(v)v' &  \tn 
% Row Count 17 (+ 4)
% Row 10
\SetRowColor{white}
FTC I Chain Rule 2 & d/dx (∫\textasciitilde{}u(x)\textasciitilde{}\textasciicircum{}v(x)\textasciicircum{} f(t)dt) & \seqsplit{f(v)v'-f(u)u'} &  \tn 
% Row Count 21 (+ 4)
% Row 11
\SetRowColor{LightBackground}
Summation Rule & d/dx \seqsplit{(Σf(x))} & Σf'(x) & The summation must be within its interval of \seqsplit{convergence} \tn 
% Row Count 27 (+ 6)
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\mymulticolumn{4}{x{8.4cm}}{a and k are constants \newline f, g, p, q, r, s, u, and v are functions of x such that f=f(x), g=g(x), p=p(x), q=q(x), r=r(x), s=s(x), u=u(x), and v=v(x), unless otherwise shown}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}----}
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% Row 1
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% Row 2
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% Row 4
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% Row Count 5 (+ 1)
% Row 5
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% Row 6
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% Row 7
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% Row 8
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\mymulticolumn{1}{x{8.4cm}}{k is a constant \newline f=f(x), g=g(x), and u=u(x) are all functions of the variable x \newline m, n ∈ ℕ\textasciitilde{}1\textasciitilde{} = \{1,2,3,4,5,...\} \newline Γ(x) is the gamma function, which defines factorials for negative non-integer numbers \newline x! = Γ(x+1) \newline {\bf{n!=n(n-1)!=n(n-1)(n-2)!=n(n-1)(n-2)(n-3)!=...}} \newline {\bf{n! = n(n-1)(n-2)(n-3)...*3*2*1}} \newline {\bf{0!=1, 1!=1}}}  \tn 
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Derivatives of Trigonometric Functions}}  \tn
% Row 0
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% Row Count 1 (+ 1)
% Row 1
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% Row 2
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% Row 3
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% Row 4
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\mymulticolumn{1}{x{8.4cm}}{d/dx (csc(x))} \tn 
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% Row 5
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% Row 6
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\mymulticolumn{1}{x{8.4cm}}{d\textasciicircum{}n\textasciicircum{}/dx\textasciicircum{}n\textasciicircum{} (sin(x)) = sin(x+nπ/2) \newline d\textasciicircum{}n\textasciicircum{}/dx\textasciicircum{}n\textasciicircum{} (cos(x)) = cos(x+nπ/2) \newline {\bf{sinh(x) = (e\textasciicircum{}x\textasciicircum{}-e\textasciicircum{}-x\textasciicircum{})/2}} \newline {\bf{cosh(x) = (e\textasciicircum{}x\textasciicircum{}+e\textasciicircum{}-x\textasciicircum{})/2}} \newline arcsinh(x) = ln(x+√(x\textasciicircum{}2\textasciicircum{}+1)) \newline arccosh(x) = ln(x+√(x\textasciicircum{}2\textasciicircum{}-1)), x≥1}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Polynomial Derivative Examples}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{d/dx (x)}} & {\bf{1}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{\bf{d/dx (x\textasciicircum{}2)}} & {\bf{2x}} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{\bf{d/dx (x\textasciicircum{}3)}} & {\bf{3x\textasciicircum{}2\textasciicircum{}}} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
d/dx (x\textasciicircum{}4) & 4x\textasciicircum{}3\textasciicircum{} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
{\bf{d/dx (1/x)}} & {\bf{-1/x\textasciicircum{}2\textasciicircum{}}} \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
{\bf{d/dx (-1/x\textasciicircum{}2\textasciicircum{})}} & {\bf{2/x\textasciicircum{}3\textasciicircum{}}} \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
d/dx (2/x\textasciicircum{}3\textasciicircum{}) & -6/x\textasciicircum{}4\textasciicircum{} \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
d/dx (-6/x\textasciicircum{}4\textasciicircum{}) & 24/x\textasciicircum{}5\textasciicircum{} \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
{\bf{d/dx (√x)}} & {\bf{1/(2√x)}} \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
{\bf{d/dx (x\textasciicircum{}1/3\textasciicircum{})}} & {\bf{1/(3x\textasciicircum{}2/3\textasciicircum{})}} \tn 
% Row Count 10 (+ 1)
% Row 10
\SetRowColor{LightBackground}
d/dx (x\textasciicircum{}1/4\textasciicircum{}) & 1/(4x\textasciicircum{}3/4\textasciicircum{}) \tn 
% Row Count 11 (+ 1)
% Row 11
\SetRowColor{white}
d/dx (x\textasciicircum{}3/2\textasciicircum{}) & 3/(2√x) \tn 
% Row Count 12 (+ 1)
% Row 12
\SetRowColor{LightBackground}
d/dx (x\textasciicircum{}5/3\textasciicircum{}) & 5/(3x\textasciicircum{}2/3\textasciicircum{}) \tn 
% Row Count 13 (+ 1)
% Row 13
\SetRowColor{white}
{\bf{d/dx (x\textasciicircum{}-√2-3\textasciicircum{})}} & {\bf{(-√2-3)x\textasciicircum{}-√2-4\textasciicircum{}}} \tn 
% Row Count 15 (+ 2)
% Row 14
\SetRowColor{LightBackground}
d/dx (1/(1+x)) & -1/(1+x)\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 16 (+ 1)
% Row 15
\SetRowColor{white}
d/dx (-1/(1+x)\textasciicircum{}2\textasciicircum{}) & 2/(1+x)\textasciicircum{}3\textasciicircum{} \tn 
% Row Count 17 (+ 1)
% Row 16
\SetRowColor{LightBackground}
d/dx (-1/(1-x)) & -1/(1-x)\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 18 (+ 1)
% Row 17
\SetRowColor{white}
d/dx (-1/(1-x)\textasciicircum{}2\textasciicircum{}) & -2/(1-x)\textasciicircum{}3\textasciicircum{} \tn 
% Row Count 19 (+ 1)
% Row 18
\SetRowColor{LightBackground}
d/dx (√(5x+1)) & 5/(2√(4x+1)) \tn 
% Row Count 20 (+ 1)
% Row 19
\SetRowColor{white}
d/dx (√(x\textasciicircum{}5\textasciicircum{}+1)) & 5x\textasciicircum{}4\textasciicircum{}/(2√(x\textasciicircum{}5\textasciicircum{}+1)) \tn 
% Row Count 21 (+ 1)
% Row 20
\SetRowColor{LightBackground}
d/dx ((2x\textasciicircum{}2\textasciicircum{}+5)\textasciicircum{}9\textasciicircum{}) & 36x(2x\textasciicircum{}2\textasciicircum{}+5)\textasciicircum{}8\textasciicircum{} \tn 
% Row Count 22 (+ 1)
% Row 21
\SetRowColor{white}
{\bf{d/dx (1)}} & {\bf{0}} \tn 
% Row Count 23 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Special/Other Derivative Examples}}  \tn
% Row 0
\SetRowColor{LightBackground}
d/dx (e\textasciicircum{}x\textasciicircum{}sin(x)) & e\textasciicircum{}x\textasciicircum{}sin(x)+e\textasciicircum{}x\textasciicircum{}cos(x) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
d/dx (e\textasciicircum{}x\textasciicircum{}cos(x)) & e\textasciicircum{}x\textasciicircum{}cos(x)-e\textasciicircum{}x\textasciicircum{}sin(x) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
d/dx (sin\textasciicircum{}x\textasciicircum{}(x)) & sin\textasciicircum{}x\textasciicircum{}(x)(ln(sin(x))+xcot(x)) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
d/dx (sin(x)\textasciicircum{}cos(x)\textasciicircum{}) & sin(x)\textasciicircum{}cos(x)\textasciicircum{}(cos\textasciicircum{}2\textasciicircum{}(x)csc(x)-sin(x)ln(sin(x))) \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
d/dx (ln(1/(1-x))) & 1/(1-x) \tn 
% Row Count 10 (+ 1)
% Row 5
\SetRowColor{white}
{\bf{d/dx (ln(x\textasciicircum{}3\textasciicircum{}+7x+12))}} & {\bf{(3x\textasciicircum{}2\textasciicircum{}+7)/(x\textasciicircum{}3\textasciicircum{}+7x+12)}} \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
d/dx (ln(e\textasciicircum{}3x\textasciicircum{}tan(x\textasciicircum{}3\textasciicircum{}))) & 3+(3x\textasciicircum{}2\textasciicircum{}sec\textasciicircum{}2\textasciicircum{}(x\textasciicircum{}3\textasciicircum{}))/(tan(x\textasciicircum{}3\textasciicircum{})) \tn 
% Row Count 14 (+ 2)
% Row 7
\SetRowColor{white}
{\bf{d/dx (1+k+t+√2+cos(a)+e+π+ln(3))}} & {\bf{0}} \tn 
% Row Count 17 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4.4 cm} x{3.6 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Trigonometric Derivative Examples}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{d/dx (-sin(x))}} & {\bf{-cos(x)}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{\bf{d/dx (-cos(x))}} & {\bf{sin(x)}} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{\bf{d/dx (sin(2x))}} & {\bf{2cos(2x)}} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
{\bf{d/dx (cos(2x))}} & {\bf{-2sin(2x)}} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
d/dx (sin\textasciicircum{}2\textasciicircum{}(x)) & 2sin(x)cos(x) \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
d/dx (cos\textasciicircum{}2\textasciicircum{}(x)) & -2cos(x)sin(x) \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
{\bf{d/dx (arctan(3x))}} & {\bf{3/(1+9x\textasciicircum{}2\textasciicircum{})}} \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
d/dx (sin(sin(x))) & cos(x)cos(sin(x)) \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
d/dx (sin(arccos(x))) & -x/√(1-x\textasciicircum{}2\textasciicircum{}) \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
{\bf{d/dx (sin(k))}} & {\bf{0}} \tn 
% Row Count 10 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.2 cm} x{4.8 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Exponential Derivative Examples}}  \tn
% Row 0
\SetRowColor{LightBackground}
d/dx (xe\textasciicircum{}x\textasciicircum{}) & e\textasciicircum{}x\textasciicircum{}+xe\textasciicircum{}x\textasciicircum{} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{\bf{d/dx (e\textasciicircum{}2x\textasciicircum{})}} & {\bf{2e\textasciicircum{}2x\textasciicircum{}}} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{\bf{d/dx (e\textasciicircum{}x$^{\textrm{2}}$\textasciicircum{})}} & {\bf{2xe\textasciicircum{}x$^{\textrm{2}}$\textasciicircum{}}} \tn 
% Row Count 4 (+ 2)
% Row 3
\SetRowColor{white}
d/dx (e\textasciicircum{}eˣ\textasciicircum{}) & e\textasciicircum{}x\textasciicircum{}e\textasciicircum{}eˣ\textasciicircum{} \tn 
% Row Count 5 (+ 1)
% Row 4
\SetRowColor{LightBackground}
{\bf{d/dx (x\textasciicircum{}x\textasciicircum{})}} & x\textasciicircum{}x\textasciicircum{}(ln(x)+1) \tn 
% Row Count 6 (+ 1)
% Row 5
\SetRowColor{white}
d/dx (2\textasciicircum{}3ˣ\textasciicircum{}) & 2\textasciicircum{}3ˣ\textasciicircum{}*3\textasciicircum{}x\textasciicircum{}*ln(2)*ln(3) \tn 
% Row Count 8 (+ 2)
% Row 6
\SetRowColor{LightBackground}
{\bf{d/dx (e\textasciicircum{}k\textasciicircum{})}} & {\bf{0}} \tn 
% Row Count 9 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{5.28 cm} x{2.72 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Logarithmic Derivative Examples}}  \tn
% Row 0
\SetRowColor{LightBackground}
d/dx (ln(1/x)) & -1/x \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
d/dx (ln(1+x)) & 1/(1+x) \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
d/dx (ln(1-x)) & -1/(1-x) \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
{\bf{d/dx (ln(x\textasciicircum{}2\textasciicircum{}))}} & {\bf{2/x}} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
d/dx (ln(x\textasciicircum{}3\textasciicircum{})) & 3/x \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
d/dx (ln(x\textasciicircum{}4\textasciicircum{})) & 4/x \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
d/dx (xln(x)) & ln(x)+1 \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
d/dx (ln(ln(x))) & 1/(xln(x)) \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
{\bf{d/dx (ln(k))}} & {\bf{0}} \tn 
% Row Count 9 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}