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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Differentiation Rules}

\begin{example}
Find \[
\left( \sqrt{x^{2}+x}\right) '.\]

\end{example}

\subsubsection*{Solution}

For this function, clearly the square root is the {}``outside{}'' function.
That is, the composition is to first take \( x \) and compute \( x^{2}+x \),
\emph{then} take the square root. So, since the chain rule tells you to work
from the outside in, first differentiate the square root. The derivative of
\[
\sqrt{u}\]
 is \[
\frac{d}{du}\sqrt{u}=\frac{1}{2\sqrt{u}},\]
so

\[
\left( \sqrt{x^{2}+x}\right) '=\frac{1}{2\sqrt{x^{2}+x}}(2x+1).\]

\end{document}
