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\centerline{Mathematics 131, section 1 -- Calculus for Physical and Life Sciences}
\centerline{Discussion 2 -- Derivative Rules}
\centerline{\it October 18, 2004}
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{\it Background}
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We have now seen 
\item{$\bullet$} the {\it product rule}:
${d\over dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x),$
\item{$\bullet$} the {\it quotient rule}:
${d\over dx}\left({f(x)\over g(x)}\right) = {f'(x)g(x) - f(x)g'(x)\over (g(x))^2},$
\item{$\bullet$} and the {\it chain rule}:
${d\over dx}(f(g(x)) = f'(g(x))g'(x)$
for derivatives.  

\noindent
Today, we want to practice using these rules
to compute some additional derivative examples, and do some 
problems using them. 
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{\it Discussion Questions}
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1)  Compute the derivative of each function with respect to the 
stated independent variable.  State which rule(s) you are using,
and simplify as much as possible.
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\item{a)}  $p(x) = {1\over \sqrt{x^2 - 6x + 9}}$
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\item{b)}  $W(s) = \left(2^s + 1\right)^4$
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\item{c)}  $g(t) = t^2 e^{-t^2}$
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\item{d)}  $f(u) =  {3^u - 1\over 3^{2u} + 1}$
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\item{e)}  $F(z) =  {\root 3 \of {e^{4z} - 1}}$
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\noindent
2)  The balance $B$ (in dollars) in a bank account earning
$1\%$ interest (``compounded continuously'') at time 
$t$ years after the initial deposit of $\$1000$ 
is $B(t) = 1000 e^{0.01 t}$.  What is the balance at $t = 5$ years?
At what rate is the balance changing at that time?  What are the units
of your second answer?  Interpret this in financial terms.
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3)  Find a formula for the derivative of a product of three functions:
$${d\over dx}(f(x)g(x)h(x)) = ?$$
Hint:  You can group the terms as $f(x)(g(x)h(x))$ and treat
the part in the parentheses as a single function.  Use your formula
to find the derivative of $H(t) = \sqrt{t}(t^2+1)^4 e^t$.
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\noindent
{\it Assignment}
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Solutions due at the end of the class period today (one per group).
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