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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{The Fundamental Theorem of Calculus}

\begin{example}
Find the derivative of \[
F(x)=\int _{0}^{x^{2}}\sqrt{1+t^{6}}dt.\]

\end{example}

\subsubsection*{Solution}

Here again, finding the integral would be hard, if not impossible. But that
is not what you are asked. If \[
G(x):=\int _{0}^{x}\sqrt{1+t^{6}}dt,\]
(note that the limits of integration are a little different), then \[
F(x)=G(x^{2}),\]
or\[
F(x)=G(u),\, \textrm{where}\, u=x^{2}.\]


The FTC we just proved shows that \[
\frac{dG}{du}=\sqrt{1+u^{6}}.\]
Don't be confused by the use of \( u \) instead of \( x \). It's actually
supposed to help. If I had used \( x \) instead of the \( u \), that would
be exactly what the FTC says. 

Then, by the chain rule, \begin{eqnarray*}
\frac{dF}{dx} & = & \frac{dG}{du}\frac{du}{dx}\\
 & = & \sqrt{1+u^{6}}2x\\
 & = & \sqrt{1+(x^{2})^{6}}2x\\
 & = & 2x\sqrt{1+x^{12}}.
\end{eqnarray*}

\end{document}
