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\author{Andrew Lorimer}
\date{}

\pagenumbering{gobble}

\begin{document}
% \begin{multicols}{2}
\renewcommand{\arraystretch}{1.5}

\hypertarget{spec---calculus}{%
\section{Spec - Calculus}\label{spec---calculus}}

\hypertarget{gradients}{%
\subsection{Gradients}\label{gradients}}

\[m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}\]

\hypertarget{limit-theorems}{%
\subsection{Limit theorems}\label{limit-theorems}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
\item
  \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
\item
  \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
\item
  \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
\end{enumerate}

\hypertarget{first-principles-derivative}{%
\subsection{First principles
derivative}\label{first-principles-derivative}}

\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]

\hypertarget{tangents-gradients}{%
\subsection{Tangents \& gradients}\label{tangents-gradients}}

\textbf{Tangent line} - defined by \(y=mx+c\) where
\(m={dy \over dx}\)\\
\textbf{Normal line} - \(\perp\) tangent
(\(m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)

\hypertarget{derivatives}{%
\subsection{Derivatives}\label{derivatives}}

\tablehead{\hline \(f(x)\) & \(f^\prime(x)\) \\ \hline \\}
\begin{supertabular}[]{@{}ll@{}}

\(kx^n\) & \(knx^{n-1}\)\tabularnewline
\(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
\(c\) & \(0\)\tabularnewline
\({u \over v}\) &
\({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
\(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
\(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
\(\sin ax\) & \(a\cos ax\)\tabularnewline
\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
\(\cos ax\) & \(-a \sin ax\)\tabularnewline
\(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
\(e^{ax}\) & \(ae^{ax}\)\tabularnewline
\(\log_e {ax}\) & \(1 \over x\)\tabularnewline
\(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline

\end{supertabular}

\hypertarget{product-rule-for-yuv}{%
\subsection{\texorpdfstring{Product rule for
\(y=uv\)}{Product rule for y=uv}}\label{product-rule-for-yuv}}

\[{dy \over dx} = u{dv \over dx} + v{du \over dx}\]

\subsection{Chain rule for $(f\circ g)$}

$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$

% Function notation:

$${\displaystyle (f\circ g)'=(f'\circ g)\cdot g'}=f'(g(x)) \cdot g'(x)$$

% $$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \text{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$}

\hypertarget{logarithms}{%
\subsection{Logarithms}\label{logarithms}}

\[\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x\]

\subsubsection{Logarithmic identities}
$\log_b (xy)=\log_b x + \log_b y$ \\
$\log_b x^n = n \log_b x$ \\
$\log_b y^{x^n} = x^n \log_b y$

\hypertarget{integration}{%
\subsection{Integration}\label{integration}}

\[\int f(x) dx = F(x) + c\]

% \begin{itemize}
% \tightlist
% \item
  area enclosed by curves
% \end{itemize}

\tablehead{\hline \(f(x)\) & \(\int f(x) \cdot dx\) \\ \hline \\}
\begin{supertabular}[]{@{}ll@{}}


\(k\) (constant) & \(kx + c\) \\
\(x^n\) & \({1 \over {n+1}}x^{n+1} + c\) \\
\(a x^{-n}\) & \(a \cdot \log_e x + c\) \\
\(e^{kx}\) & \({1 \over k} e^{kx} + c\) \\
\(e^k\) & \(e^kx + c\) \\
\(\sin kx\) & \(-{1 \over k} \cos (kx) + c\) \\
\(\cos kx\) & \({1 \over k} \sin (kx) + c\) \\
\({f^\prime (x)} \over {f(x)}\) & \(\log_e f(x) + c\) \\
\(g^\prime(x)\cdot f^\prime(g(x)\) & \(f(g(x))\) (chain rule) \\
\(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\) \\
\({1 \over {ax+b}}\) & \({1 \over a} \log_e (ax+b) + c\) \\
\((ax+b)^n\) & \({1 \over {a(n+1)}}(ax+b)^{n-1} + c\) \\

\end{supertabular}

\hypertarget{definite-integrals}{%
\subsection{Definite integrals}\label{definite-integrals}}

\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}\]

\hypertarget{kinematics}{%
\subsection{Kinematics}\label{kinematics}}

\textbf{position \(x\)} - distance from origin or fixed point\\
\textbf{displacement \(s\)} - change in $x$ from starting point \\
\textbf{velocity \(v\)} - change in position with respect to time\\
\textbf{acceleration \(a\)} - change in velocity\\
\textbf{speed} - magnitude of velocity

\large{
\tablehead{}
\begin{supertabular}[]{@{}ll@{}}
% \toprule
& no\tabularnewline
\(v=u+at\) & \(s\)\tabularnewline
\(s=ut + {1 \over 2} at^2\) & \(v\)\tabularnewline
\(v^2 = u^2 + 2as\) & \(t\)\tabularnewline
\(s= {1 \over 2}(u+v)t\) & \(a\)\tabularnewline



\end{supertabular}
}
% \end{multicols}
\end{document}
