\subsection*{Derivatives}

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\begin{tabularx}{\columnwidth}{rX}
  \hline
  \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
  \hline
  \(\sin x\) & \(\cos x\)\\
  \(\sin ax\) & \(a\cos ax\)\\
  \(\cos x\) & \(-\sin x\)\\
  \(\cos ax\) & \(-a \sin ax\)\\
  \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
  \(e^x\) & \(e^x\)\\
  \(e^{ax}\) & \(ae^{ax}\)\\
  \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
  \(\log_e x\) & \(\dfrac{1}{x}\)\\
  \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
  \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
  \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
  \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
  \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
  \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
  \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
  \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
  \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
  \hline
\end{tabularx}

\vfill
\vtop to 5cm {
  \flushbottom
  \subsubsection*{Index identities}
  \begin{align*}
    b^{m+n} &= b^m \cdot b^n \\
    (b^m)^n &= b^{m \cdot n} \\
    (b \cdot c)^n  &=  b^n \cdot c^n \\
    {a^m \div a^n}  &=  {a^{m-n}}
  \end{align*}
}
  

\subsection*{Antiderivatives}

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\begin{tabularx}{\columnwidth}{rX}
  \hline
  \(f(x)\) & \(\int f(x) \cdot dx\) \\
  \hline
  \(k\) (constant) & \(kx + c\)\\
  \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
  \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
  \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
  \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
  \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
  \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
  \(e^k\) & \(e^kx + c\)\\
  \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
  \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
  \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
  \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
  \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
  \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
  \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
  \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
  \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
  \hline
\end{tabularx}
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\vspace{1em}
Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\)

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\vtop to 5cm {
  \flushbottom
  \subsubsection*{Logarithmic identities}
  \begin{align*}
    \log_b (xy) &= \log_b x + \log_b y \\
    \log_b\left(\frac{x}{y}\right) &= \log_b(x) - \log_b(y) \\
    \log_b x^n &= n \log_b x \\
    \log_b y^{x^n} &= x^n \log_b y
  \end{align*}
}
