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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\paragraph*{Inverse Functions}


\subsection*{Derivatives of inverse functions; implicit differentiation}

\begin{example}
Use implicit differentiation to find the derivative of \[
\sqrt{x}.\]

\end{example}

\subsubsection*{Solution}

Start off with the {}``frontwards{}'' function, \( f(x)=x^{2} \). Then the
square root is the inverse, \[
f^{-1}(x)=\sqrt{x}.\]
Following the method (not just plugging into the formula) we just went through
in general, \begin{eqnarray*}
f^{-1}(x)=y & = & \sqrt{x}\\
y^{2} & = & x
\end{eqnarray*}
Differentiating both sides,\begin{eqnarray*}
y^{2} & = & x\\
2y\frac{dy}{dx} & = & 1,\: \textrm{or}\\
\frac{dy}{dx} & = & \frac{1}{2y}\\
 & = & \frac{1}{2\sqrt{x}}.
\end{eqnarray*}
Of course, we knew that already, but this is a new way to approach it. 
\end{document}
