sum rule
SumRule
2002-02-24 22:03:53
2004-03-22 14:20:58
Theorem
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\newcommand{\D}[1]{\ensuremath{\mathrm{d}#1}}
\newcommand{\DDX}{\ensuremath{\frac{\D{}}{\D{x}}}}
\PMlinkescapeword{states}
The \emph{sum rule} states that
\begin{equation*}
\DDX\left[f(x)+g(x)\right] = f'(x) + g'(x)
\end{equation*}
\subsubsection*{Proof}
See the \PMlinkname{proof of the sum rule}{ProofOfSumRule}.
\subsubsection*{Examples}
\begin{eqnarray*}
\DDX(x + 1) & = & \DDX x + \DDX 1 = 1 \\
\DDX(x^2 - 3x + 2) & = & \DDX x^2 + \DDX(-3x) + \DDX(2) = 2x-3 \\
\DDX(\sin x + \cos x) & = & \DDX\sin x + \DDX\cos x = \cos x - \sin x
\end{eqnarray*}