1\subsection*{Derivatives}
2\rowcolors{1}{white}{peach}
4\renewcommand{\arraystretch}{1.4}
5\begin{tabularx}{\columnwidth}{rX}
7 \hline
8 \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
9 \hline
10 \(\sin x\) & \(\cos x\)\\
11 \(\sin ax\) & \(a\cos ax\)\\
12 \(\cos x\) & \(-\sin x\)\\
13 \(\cos ax\) & \(-a \sin ax\)\\
14 \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
15 \(e^x\) & \(e^x\)\\
16 \(e^{ax}\) & \(ae^{ax}\)\\
17 \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
18 \(\log_e x\) & \(\dfrac{1}{x}\)\\
19 \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
20 \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
21 \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
22 \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
23 \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
24 \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
25 \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
26 \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
27 \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
28 \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
29 \hline
30\end{tabularx}
31\vfill
33\vtop to 5cm {
34 \flushbottom
35 \subsubsection*{Index identities}
36 \begin{align*}
37 b^{m+n} &= b^m \cdot b^n \\
38 (b^m)^n &= b^{m \cdot n} \\
39 (b \cdot c)^n &= b^n \cdot c^n \\
40 {a^m \div a^n} &= {a^{m-n}}
41 \end{align*}
42}
43
44\subsection*{Antiderivatives}
46\rowcolors{1}{white}{lblue}
48\renewcommand{\arraystretch}{1.4}
49\begin{tabularx}{\columnwidth}{rX}
51 \hline
52 \(f(x)\) & \(\int f(x) \cdot dx\) \\
53 \hline
54 \(k\) (constant) & \(kx + c\)\\
55 \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
56 \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
57 \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
58 \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
59 \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
60 \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
61 \(e^k\) & \(e^kx + c\)\\
62 \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
63 \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
64 \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
65 \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
66 \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
67 \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
68 \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
69 \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
70 \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
71 \hline
72\end{tabularx}
73\rowcolors{2}{white}{white}
74\vspace{1em}
76Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\)
77\vfill
79\vtop to 5cm {
80 \flushbottom
81 \subsubsection*{Logarithmic identities}
82 \begin{align*}
83 \log_b (xy) &= \log_b x + \log_b y \\
84 \log_b\left(\frac{x}{y}\right) &= \log_b(x) - \log_b(y) \\
85 \log_b x^n &= n \log_b x \\
86 \log_b y^{x^n} &= x^n \log_b y
87 \end{align*}
88}