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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{The Derivative of a Function, Continued}
\centerline{October 17, 2017}
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\noindent
{\it Background}
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\noindent
We will now concentrate on finding \emph{general formulas} for derivatives.
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{\it Questions}
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\begin{enumerate}
\item[(0)] Use the definition to compute $f'(a)$ if $f$ is the linear function $f(x) = 5x + 3$.
Does your answer make sense in terms of $f'(a)$ as the slope of the tangent line?
\item[(1)] Compute $f'(a)$--the derivative at
a general $x = a$ for $f(x) = x^2$.
\item[(2)] Compute $f'(a)$--the derivative at
a general $x = a$ for $f(x) = x^3$.
\item[(3)] Compute $f'(a)$--the derivative at
a general $x = a$ for $f(x) = \sqrt{x}$.  Here there is a restriction on which $a$
``work.''  What is that restriction?  Does this make sense, thinking of the graph $y = \sqrt{x}$?
(Note:  this is part of the parabola with equation $x = y^2$.)
\item[(4)] Compute $f'(a)$--the derivative at
a general $x = a$ for $f(x) = \frac{1}{x}$.  Does your formula make sense, thinking of 
the shape of the graph $y = \frac{1}{x}$?  In particular, what is true about $f'(a)$ if 
$a$ is very close to zero?  And what about $a$ very large in absolute value?
\item[(5)]  Can you see a general pattern in all of these examples?  Think about writing 
$\sqrt{x}$ and $\frac{1}{x}$ as powers.  Do they fit the same pattern?
\end{enumerate}
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