Andrew's git - notes.git/blob - spec/calculus-rules.tex

   1\subsection*{Derivatives}
   2
   3\rowcolors{1}{white}{peach}
   4\renewcommand{\arraystretch}{1.4}
   5
   6\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
  32\vfill
  33\vtop to 5cm {
  34  \flushbottom
  35  \subsubsection*{Index identities}
  36  \begin{align*}
  37    a^{x+y} &= a^x \cdot a^y \\
  38    a^{x-y} &= a^x \div a^y \\
  39    (a^x)^y &= a^{x \cdot y} \\
  40    (a \cdot b)^x  &=  a^x \cdot b^x
  41  \end{align*}
  42}
  43  
  44
  45\subsection*{Antiderivatives}
  46
  47\rowcolors{1}{white}{lblue}
  48\renewcommand{\arraystretch}{1.4}
  49
  50\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
  75\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
  78\vfill
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  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 y^{x^n} &= x^n \log_b y \\
  86    \log_b x^n &= n \log_b x
  87  \end{align*}
  88}