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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{The Derivative Function}
\centerline{October 20, 2015}
\vskip 10pt
\noindent
{\it Background}
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Recall that last time, we introduced the \emph{derivative} of a function 
$f$ at $x$:
$$f'(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h},$$
provided the limit exists. 

In today's video we introduced the idea of deriving formulas for $f'(x)$ for a general x, and
considering $f'$ as a new function in its own right. We said:
\begin{enumerate}
\item[(1)]  If $f(x) = x^n$ for any number $n$, then $f'(x) = n x^{n-1}$.
\item[(2)]  If $f'(x)$ exists, then so does $(kf)'(x)$ and $(kf)'(x) = k f'(x)$.  
\item[(3)]  If $f'(x)$ and $g'(x)$ both exist, then so does $(f + g)'(x)$, and 
$(f + g)'(x) = f'(x) + g'(x)$.  
\end{enumerate}
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\noindent
{\it Questions}
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\noindent
\begin{enumerate}
\item[(1)]  Compute $f'(x)$ for $f(x) = x^{5/3}$ using point (1) in the Background above.
\item[(2)]  Compute $f'(x)$ for $f(x) = x^4 + 5x^2 + 7 + 2x^{-3}$ using points (1), (2), and (3)
in the Background above.
\item[(3)]  Make up your own example function of the same type as the one in (2) and
compute $f'(x)$ using points (1), (2), and (3) in the Background.
\item[(4)]  What is $f'(x)$ if $f(x) = \pi^\pi$?  (Be careful!) 
\end{enumerate}
\noindent
The video for today gave the idea for showing point (1) in the Background when 
$n$ is a positive integer, and stated that the same formula works for all $n$.
Let's verify this in some special cases.
\begin{enumerate}
\item[(5)] Verify the formula in point (1) for $f(x) = x^4$.  That is, you want to show
$f'(x) = 4 x^3$ using the definition of $f'(x)$.
\item[(6)] Verify the formula in point (1) for $f(x) = \sqrt{x} = x^{1/2}$.  That is, you want to show
$f'(x) = \frac{1}{2} x^{-1/2}$ using the definition of $f'(x)$.
\item[(7)] Verify the formula in point (1) for $f(x) = \frac{1}{x^2} = x^{-2}$.  
That is, you want to show $f'(x) = -2 x^{-3}$ using the definition of $f'(x)$.
\item[(8)]  Why does point (2) in the background work?  Explain in terms of the 
limit definition, and in terms of how the graphs $y = f(x)$ and $y = kf(x)$ are
related. 
\end{enumerate}

Finally, it is important to realize that the limit in the definition of $f'(x)$ can fail
to exist at some of the $x$ in the domain of $f$.  
\begin{enumerate}
\item[(9)]  Consider $y = |x|$ (the absolute value function).  Does 
$$f'(0) = \lim_{h\to 0} \frac{|h| - 0}{h}$$
exist?  Why or why not?   Does this make sense in terms of the graph $y = |x|$?
\end{enumerate}
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