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\begin{center}{\bf \large Partial Derivatives for Math 305}\end{center}


Since partial derivates are not covered in Calculus II (Math 250), but the textbook for Math 305
assumes you have had this, I made this handout. If you have had Calculus III (Math 251) or 
have seen partial derivates elsewhere you shouldn't need to read this. 

Let $f(x,y)$ be a function of two variables. The {\bf partial derivative}
of $f$ with respect to $x$ measures the rate of change in the value
of $f(x,y)$ as $x$ varies, but $y$ is held fixed. There are several common 
notations for this:
\[
\frac{\bd f}{\bd x}  \hspace{.3in} \bd_x f  \hspace{.3in} f_x
\]
all mean the partial derivative of $f$ with respect to $x$.


In pratice this is easy 
to compute, you pretend $y$ is a constant and find the derivate in usual
way. Here are some examples.

\begin{enumerate}

\item Let $f(x,y) = x^2y^3$. Then  $\dis\frac{\bd f}{\bd x} = 2xy^3$.

\item Let $f(x,y) = x \sin xy$. Then  $\dis\frac{\bd f}{\bd x} = \sin xy + xy \cos xy$. Here we used the
product rule and the chain rule. If you are confused, replace $y$ with the number $3$ or the letter $a$
and compute the derivative as usual. 

\item Let $g(t,z) = \tan t^2z + \sinh z^3$. Then  $\dis\frac{\bd g}{\bd t} = 2tz \sec^2 t^2z$. 

\end{enumerate}

The partial derivate of $f(x,y)$ with respect to $y$ is defined similary as the rate of change of $f$
as $y$ varies and $x$ is held fixed.  Here are some examples.

\begin{enumerate}

\item [4.] Let $f(x,y) = x^2y^3$. Then $\dis\frac{\bd f}{\bd y} = 3x^2y^2$.

\item [5.] Let $f(x,y) = x \sin xy$. Then $\dis\frac{\bd f}{\bd y} = x^2 \cos xy$. Only the chain rule was needed.

\item [6.] Let $g(t,z) = \tan t^2z + \sinh z^3$. Then  $\dis\frac{\bd g}{\bd z} = t^2 \sec^2 t^2z + 3z^2 \cosh z^3$. 

\end{enumerate}


You can refer to Section 11.3 of the calculus textbook used at SIU for more. Any calculus textbook will have a section on 
partial derivatives. 

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