%This command provides the mathematic (calculus) of the second
%column of the page 6.
%
%This command has one parameter:
%       1) The width of the math text
\newcommand\TSixCalculusOne[1]{%
   \def\dudx{\unskip\,\frac{du}{dx}}
   \parbox[t]{#1}{%
      \TSixColTwoFontSize
      \deflength{\VSpace}{1.15\TSixSkipFormulae} %Useful to add more space for the derivatives.
      \TSixTitle{Derivatives:}
      \AdjustSpace{1ex plus .5ex minus .5ex}
      \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\VSpace}{\BigChar}{\StyleBold}
      \Fm{\frac{d(cu)}{dx} = c \dudx}
      \Fm{\frac{d(u+v)}{dx} = \dudx + \frac{dv}{dx}}
      \Fm{\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \dudx}
      \Fm{\frac{d(u^n)}{dx} = nu^{n-1}\dudx}
      \Fm{\frac{d(u/v)}{dx} = \frac{v\big(\dudx\big) - u\big(\frac{dv}{dx}\big)}{v^2}}
      \Fm{\frac{d(e^{cu})}{dx} = ce^{cu}\dudx}
      \Fm{\frac{d(c^u)}{dx} = (\ln c) c^u\dudx}
      \Fm{\frac{d(\ln u)}{dx} = \frac{1}{u} \dudx}
      \Fm{\frac{d(\sin u)}{dx} = \cos u \dudx}
      \Fm{\frac{d(\cos u)}{dx} = - \sin u \dudx}
      \Fm{\frac{d(\tan u)}{dx} = \sec^2 u \dudx}
      \Fm{\frac{d(\cot u)}{dx} = \csc^2 u \dudx}
      \Fm{\frac{d(\sec u)}{dx} = \tan u \,\sec u \dudx}
      \Fm{\frac{d(\csc u)}{dx} = - \cot u \,\csc u \dudx}
      \Fm{\frac{d(\arcsin u)}{dx} = \frac{1}{\sqrt{1 - u^2}}\dudx}
      \Fm{\frac{d(\arccos u)}{dx} = \frac{-1}{\sqrt{1 - u^2}}\dudx}
      \Fm{\frac{d(\arctan u)}{dx} = \frac{1}{1 + u^2}\dudx}
      \Fm{\frac{d(\arccot u)}{dx} = \frac{-1}{1 + u^2}\dudx}
      \Fm{\frac{d(\arcsec u)}{dx} = \frac{1}{u \sqrt{1 - u^2}}\dudx}
      \Fm{\frac{d(\arccsc u)}{dx} = \frac{-1}{u \sqrt{1 - u^2}}\dudx}
      \Fm{\frac{d(\sinh u)}{dx} = \cosh u \dudx}
      \Fm{\frac{d(\cosh u)}{dx} = \sinh u \dudx}
      \Fm{\frac{d(\tanh u)}{dx} = \sech^2 u \dudx}
      \Fm{\frac{d(\coth u)}{dx} = -\csch^2 u \dudx}
      \Fm{\frac{d(\sech u)}{dx} = -\sech u \,\tanh u\dudx}
      \Fm{\frac{d(\csch u)}{dx} = -\csch u \,\coth u\dudx}
      \Fm{\frac{d(\arcsinh u)}{dx} = \frac{1}{\sqrt{1 + u^2}}\dudx}
      \Fm{\frac{d(\arccosh u)}{dx} = \frac{1}{\sqrt{u^2 - 1}}\dudx}
      \Fm{\frac{d(\arctanh u)}{dx} = \frac{1}{1 - u^2}\dudx}
      \Fm{\frac{d(\arccoth u)}{dx} = \frac{1}{u^2 - 1}\dudx}
      \Fm{\frac{d(\arcsech u)}{dx} = \frac{-1}{u \sqrt{1 - u^2}}\dudx} 
      \Fm{\frac{d(\arccsch u)}{dx} = \frac{-1}{\vert u \vert \sqrt{1 + u^2}}\dudx.}
      \end{DisplayFormulae}

      \TSixTitle{Integrals:}
      \begin{DisplayFormulae}{1}{\SpaceBeforeFormula}{\TSixSkipFormulae}{\BigChar}{\StyleBold}
      \Fm{\int c u \dx = c \int u \dx}
      \Fm{\int (u + v) \dx = \int u \dx + \int v \dx}
      \Fm{\int x^n \dx = \frac{1}{n+1}x^{n+1}
          \MathRemark{n \neq -1}}
      \Fm{\int \frac{1}{x} dx = \ln \ValAbs{x}}
      \Fm{\int e^x \dx = e^x}
      \Fm{\int \frac{dx}{1 + x^2} = \arctan x}
      \Fm{\int u \frac{dv}{dx} dx = uv - \int v {du \over dx} dx}
      \Fm{\int \sin \xdx = -\cos x}
      \Fm{\int \cos \xdx = \sin x}
      \Fm{\int \tan \xdx = -\ln \ValAbs{\cos x}}
      \Fm{\int \cot \xdx = \ln \ValAbs{\cos x}}
      \Fm{\int \sec \xdx = \ln \ValAbs{\sec x + \tan x}}
      \Fm{\int \csc \xdx = \ln \ValAbs{\csc x + \cot x}}
      \Fm{\int \arcsin \tfrac{x}{a} dx = \arcsin \tfrac{x}{a} + \sqrt{a^2 - x^2}
          \MathRemark{a > 0}}
      \end{DisplayFormulae}
   }
}
