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\begin{document}

{\Large Partial derivatives}

\begin{definition}
If $f(x,y)$ is a function of two variables, its partial 
derivative with respect to $x$, \pz denoted either
$\frac{\partial f}{\partial x}$ or $f_x(x,y)$, \pz is the function 
obtained by treating $y$ as a constant and differentiating
with respect to $x$. \pz  Similarly, the partial derivative
with respect to $y$, denoted $\frac{\partial f}{\partial y}$ or $f_y$,
is obtained by treating $x$ as a constant and differentiating
with respect to $y$. \pb
\end{definition}

So, the game is to treat one variable as a constant while
differentiating with respect to the other variable. \pz

\begin{example}
Find the partial derivatives of the following functions: \pz
\begin{itemize}
\item $f(x,y) = x^3 y^2 - 7 xy^3 + \sqrt{5} x^2 + 5$. \pz
\item $f(x,y) = x^2 e^{\sqrt{xy}}$ \pz
\end{itemize}
\end{example}

\pb

{\large Geometric and practical interpretations} \pz

Recall that one of the first and most important
interpretations of the derivative was that it was the slope
of the tangent line to a curve. \pz  There is a similar
first interpretation of the partial derivative. \pz
As we illustrate on the overhead, the partial derivative
with respect to $x$ is the slope, within the plane where
$y$ is fixed at some $c$ and $x$ and $z$ are allowed to vary,
of the {\em tangent line to the graph of the function}. \pz

Thus, practically speaking, if you were walking along some
surface which you could then think of as the graph of a two-variable
function, \pz then the partial derivatives tell you about how steep your 
climb or descent will be if you walk parallel to the $x$ or $y$ axes
(due north-south or east-west).  \pz  You might wonder how you could
find out about the steepness of the climb or descent if you travel
northeast \pz - that's the topic of the {\em gradient} of a function. \pb

{\large Higher-order partial derivatives} \pz

As in the case of a single variable, we are free to take
a derivative of a derivative. \pz   The notation works as follows:
\begin{definition}
The partial derivative with respect to $x$ of the partial with respect to 
$x$ is  $f_{xx}$ or $\frac{\partial^2 f}{\partial x^2}$. \\ \pz
The partial derivative with respect to $y$ of the partial with respect to
$x$ is $f_{xy}$ or $\frac{\partial^2 f}{\partial y \partial x}$. \\ \pz
The partial derivative with respect to $x$ of the partial with respect to 
$y$ is  $f_{yx}$ or $\frac{\partial^2 f}{\partial x \partial y}$. \\ \pz
The partial derivative with respect to $y$ of the partial with respect to 
$y$ is  $f_{yy}$ or $\frac{\partial^2 f}{\partial y^2}$. \\ \pz
\end{definition}

\begin{example}
Find all four second-order partial derivatives of $f(x,y) = x^2 y^3 + e^{xy}$ \pz
and $f(x,y) = \ln(x + y)$ \pz
\end{example}
\pb


\end{document}
 