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%MODULE: Continuity
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\begin_layout Title
Derivatives of polynomials 
\end_layout

\begin_layout Subtitle
In this module we will determine more efficient ways of computing the derivative
s of 
\emph on
polynomials
\emph default
.
 We use the precise definition of the derivative to explain these methods.
 
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%1
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{
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Defining the problem
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}
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\begin_layout Standard
\noindent
Recall the definition of the derivative as a function.
\end_layout

\begin_layout Definition
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[
\end_layout

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(The derivative as a function)
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]
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\end_inset

If a function 
\begin_inset Formula $\ec f$
\end_inset

 is differentiable at every point in its domain, then 
\emph on

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\backslash
hilite
\end_layout

\end_inset

 the derivative of 
\begin_inset Formula $\ec f$
\end_inset


\emph default
 is
\begin_inset Formula 
\[
{\ec f'(x)=\lim_{x\rightarrow a}\frac{f(x+h)-f(x)}{h}}.
\]

\end_inset

 It is tedious to compute the derivative using the precise definition.
 Polynomials have a certain structure that is different from trigonometric,
 exponential, or logarithmic functions.
 We'd like to exploit that structure and find a shorter way of computing
 the derivative for polynomials.
 
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%2
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{
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SAGE worksheets
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}
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\begin_layout Standard
\noindent
You will need a blank SAGE worksheet for this module.
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{
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The derivative of a polynomial
\begin_inset ERT
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}
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\end_layout

\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f(x)$
\end_inset

 be a polynomial.
 We want to compute the derivative of 
\begin_inset Formula $\ec f(x)$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
The simplest form of a polynomial is a monomial 
\begin_inset Formula $\ec x^{n}$
\end_inset

, where 
\begin_inset Formula $\ec n$
\end_inset

 is a positive integer.
 Before looking at this 
\begin_inset Quotes eld
\end_inset

generic
\begin_inset Quotes erd
\end_inset

 monomial, we'll look at a few examples: 
\begin_inset Formula $\ec x$
\end_inset

, 
\begin_inset Formula $\ec x^{2}$
\end_inset

, and 
\begin_inset Formula $\ec x^{3}$
\end_inset

.
\end_layout

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{
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The derivative of 
\begin_inset Formula $x$
\end_inset


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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)={\hilite x}$
\end_inset

 is
\begin_inset Formula 
\[
\ec\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow0}\frac{(\cancel{x}+h)-\cancel{x}}{h}=\lim_{h\rightarrow0}1.
\]

\end_inset

 Hence we can say
\end_layout

\begin_layout Theorem
\noindent
 
\end_layout

\begin_deeper
\begin_layout Itemize
If 
\begin_inset Formula $\ec f(x)=x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x)=1$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
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{
\end_layout

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The derivative of 
\begin_inset Formula $x^{2}$
\end_inset


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}
\end_layout

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<1>
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\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)={\hilite x^{2}}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\ec\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \ec=\lim_{h\rightarrow0}\frac{(x+h)^{2}-x^{2}}{h}
\end{align*}

\end_inset


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<2>
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\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}\frac{(x+h)^{2}-x^{2}}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{(\cancel{x^{2}}+2xh+h^{2})-\cancel{x^{2}}}{h}
\end{align*}

\end_inset


\end_layout

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\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}\frac{(\cancel{x^{2}}+2xh+h^{2})-\cancel{x^{2}}}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{\cancel{h}(2x+h)}{\cancel{h}}
\end{align*}

\end_inset

 
\begin_inset ERT
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\backslash
begin{explanation}
\end_layout

\end_inset

Factored a common 
\begin_inset Formula $h$
\end_inset

 in the numerator.
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\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

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\begin_layout Pause

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<4>
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\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}\frac{\cancel{h}(2x+h)}{\cancel{h}}\\
 & \ec=\lim_{h\rightarrow0}2x+\cancelto{0}{h}.
\end{align*}

\end_inset

 
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begin{explanation}
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\end_inset

Evaluated the limit be substituting 
\begin_inset Formula $h=0$
\end_inset

.
\begin_inset Newline newline
\end_inset

 We can do this because 
\begin_inset Formula $2x+h$
\end_inset

 is a continuous function.
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\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

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\begin_layout Pause

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\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}2x+\cancelto{0}{h}\\
 & \ec=2x.
\end{align*}

\end_inset

 Hence we can say
\end_layout

\begin_layout Theorem
\noindent
 
\end_layout

\begin_deeper
\begin_layout Itemize
If 
\begin_inset Formula $\ec f(x)=x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x)=1$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

, then 
\begin_inset Formula $\ec f'(x)=2x$
\end_inset

.
 
\end_layout

\end_deeper
\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
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{
\end_layout

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The derivative of 
\begin_inset Formula $x^{3}$
\end_inset


\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

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{
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\end_layout

\end_inset


\end_layout

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<1>
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\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)={\hilite x^{3}}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\ec\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \ec=\lim_{h\rightarrow0}\frac{(x+h)^{3}-x^{3}}{h}
\end{align*}

\end_inset


\end_layout

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<2>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{3}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}\frac{(x+h)^{3}-x^{3}}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{(\cancel{x^{3}}+3x^{2}h+3xh^{2}+h^{3})-\cancel{x^{3}}}{h}
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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\begin_inset ERT
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<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{3}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}\frac{(\cancel{x^{3}}+3x^{2}h+3xh^{2}+h^{3})-\cancel{x^{3}}}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{\cancel{h}(3x^{2}+3xh+h^{2})}{\cancel{h}}
\end{align*}

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Factored a common 
\begin_inset Formula $h$
\end_inset

 in the numerator.
\begin_inset ERT
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\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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\begin_inset ERT
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<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{3}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}\frac{\cancel{h}(3x^{2}+3xh+h^{2})}{\cancel{h}}\\
 & \ec=\lim_{h\rightarrow0}3x^{2}+3x\cancelto{0}{h}+\cancelto{0}{h^{2}}
\end{align*}

\end_inset

 
\begin_inset ERT
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\backslash
begin{explanation}
\end_layout

\end_inset

Evaluated the limit be substituting 
\begin_inset Formula $h=0$
\end_inset

.
\begin_inset Newline newline
\end_inset

 We can do this because 
\begin_inset Formula $3x^{2}+3xh+h$
\end_inset

 is a continuous function.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Using the precise definition, the derivative of 
\begin_inset Formula $\ec f(x)=x^{3}$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\dims\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} & \dims=\lim_{h\rightarrow0}3x^{2}+3x\cancelto{0}{h}+\cancelto{0}{h^{2}}\\
 & \ec=3x^{2}.
\end{align*}

\end_inset

 Hence we can say
\end_layout

\begin_layout Theorem

\end_layout

\begin_deeper
\begin_layout Itemize
If 
\begin_inset Formula $\ec f(x)=x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x)=1$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

, then 
\begin_inset Formula $\ec f'(x)=2x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec f(x)=x^{3}$
\end_inset

, then 
\begin_inset Formula $\ec f'(x)=3x^{2}$
\end_inset

.
 
\end_layout

\end_deeper
\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
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{
\end_layout

\end_inset

Geometrically
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

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{
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
You can see this geometrically in SAGE.
 Plot 
\begin_inset Formula $\ec f(x)$
\end_inset

 and 
\begin_inset Formula $\ec f'(x)$
\end_inset

 on the same graph, and observe how the graph of 
\begin_inset Formula $\ec f(x)$
\end_inset

 relates to the graph of 
\begin_inset Formula $\ec f'(x)$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

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<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
When 
\begin_inset Formula $\ec f(x)=x$
\end_inset

, 
\begin_inset Formula $\ec f'(x)=1$
\end_inset

.
 This reflects the constant slope of the tangent line.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_x_1.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
When 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

, 
\begin_inset Formula $\ec f'(x)=2x$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_x2_2x.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Notice that when the slope of the line tangent to 
\begin_inset Formula $\ec x^{2}$
\end_inset

 is negative---as it is for 
\begin_inset Formula $\ec x\in(-\infty,0)$
\end_inset

---the 
\begin_inset Formula $y$
\end_inset

-values of 
\begin_inset Formula $\ec f'(x)$
\end_inset

 are negative.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
When 
\begin_inset Formula $\ec f(x)=x^{3}$
\end_inset

, 
\begin_inset Formula $\ec f'(x)=3x^{2}$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_x3_3x2.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Notice that the 
\begin_inset Formula $y$
\end_inset

-values of 
\begin_inset Formula $\ec f'(x)$
\end_inset

 are always positive in this graph, reflecting the fact that the slope of
 the line tangent to 
\begin_inset Formula $\ec f(x)$
\end_inset

 is always positive.
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Did you notice a pattern?
\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
You may have noticed the outlines of a pattern here: 
\end_layout

\begin_layout Standard
\align center
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{
\end_layout

\end_inset


\begin_inset ERT
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\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="2">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f(x)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f'(x)$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1=1x^{0}$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x^{2}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2x$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x^{3}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3x^{2}$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x^{4}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
?? 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Can you guess what the next derivative should be?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
It looks as if the derivative of 
\begin_inset Formula $\ec f(x)=x^{4}$
\end_inset

 should be 
\begin_inset Formula $\ec f'(x)=4x^{3}$
\end_inset

.
 You might guess further that the derivative of 
\begin_inset Formula $\ec f(x)=x^{n}$
\end_inset

 is 
\begin_inset Formula $\ec f'(x)=nx^{n-1}$
\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A pattern in the precise definition
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
To prove this guess, look back at the algebra of the precise definition
 of the derivative.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Everything is the same each time except
\begin_inset Formula 
\begin{align*}
\ec(x+h) & \ec=x^{1}+h\\
\ec(x+h)^{2} & \ec=x^{2}+2xh+h^{2}\\
\ec(x+h)^{3} & \ec=x^{3}+3x^{2}h+3xh^{2}+h^{3}
\end{align*}

\end_inset

Notice a pattern here: 
\begin_inset Formula $\ec(x+h)^{n}=x^{n}+h(nx^{n-1}+g)$
\end_inset

 where 
\begin_inset Formula $\ec g$
\end_inset

 is a polynomial with a power of 
\begin_inset Formula $\ec h$
\end_inset

 
\emph on
in every term
\emph default
.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Test this pattern using SAGE: for different values of 
\begin_inset Formula $\ec n$
\end_inset

, expand 
\begin_inset Formula $\ec(x+h)^{n}$
\end_inset

 and verify the result looks like 
\begin_inset Formula $\ec x^{n}+h(nx^{n-1}+g)$
\end_inset

.
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
(Hint 1: Use the 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 expand
\family default
 command to expand each product.
 For example, the command 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 expand((x+h)‸4)
\family default
 will expand the product 
\begin_inset Formula $\ec(x+h)^{4}$
\end_inset

.)
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
(Hint 2: You can see as many examples as you want using a 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 for
\family default
 loop.
 For example, 
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace -2ex
\end_inset


\begin_inset Box Frameless
position "c"
hor_pos "c"
has_inner_box 1
inner_pos "c"
use_parbox 1
use_makebox 0
width "4in"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout

\family typewriter
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 var('h')
\begin_inset Newline newline
\end_inset

 for n in range(19):
\begin_inset Newline newline
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
phantom
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

expand((x+h)‸(n+2)) 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset VSpace -2ex
\end_inset

will compute 
\begin_inset Formula $\ec(x+h)^{2}$
\end_inset

, 
\begin_inset Formula $\ec(x+h)^{3}$
\end_inset

, \SpecialChar \ldots{}
 
\begin_inset Formula $\ec(x+h)^{20}$
\end_inset

.)
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
Try to think about why this pattern is true for every positive integer 
\begin_inset Formula $\ec n$
\end_inset

.
 We will not dwell on a proof, but your instructor and/or textbook can provide
 one if you ask.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The derivative of a monomial
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We can now prove
\end_layout

\begin_layout Theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(The derivative of a monomial)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

If 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec f(x)=x^{n}$
\end_inset

 for some 
\begin_inset Formula $\ec n\in\mathbb{N}$
\end_inset

, 
\end_layout

\begin_layout Standard
then 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f'(x)=nx^{n-1}$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Standard
Why?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\ec f'(x) & \ec=\lim_{h\rightarrow0}\frac{(x+h)^{n}-x^{n}}{h}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

 From the precise definition of a derivative.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{(x+h)^{n}-x^{n}}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{(\cancel{x^{n}}+h(nx^{n-1}+g))-\cancel{x^{n}}}{h}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

 From the pattern described previously.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{(\cancel{x^{n}}+h(nx^{n-1}+g)-\cancel{x^{n}}}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{\cancel{h}(nx^{n-1}+g)}{\cancel{h}}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

 It is important to note that since 
\begin_inset Formula $h\rightarrow0$
\end_inset

 we can assume that 
\begin_inset Formula $h\neq0$
\end_inset

.
\begin_inset Newline newline
\end_inset

 Hence this division is perfectly legal.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\cancel{h}(nx^{n-1}+g)}{\cancel{h}}\\
 & \ec=\lim_{h\rightarrow0}nx^{n-1}+\cancelto{0}{g}\\
 & \ec=nx^{n-1}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset


\begin_inset Formula $g$
\end_inset

 is a polynomial with a power of 
\begin_inset Formula $h$
\end_inset

 
\emph on
in every term
\emph default
, and 
\begin_inset Formula $h\rightarrow0$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We can now say, 
\emph on
without using the precise definition
\emph default
, that 
\end_layout

\begin_layout Itemize
if 
\begin_inset Formula $\ec f(x)=x^{27}$
\end_inset

, 
\end_layout

\begin_layout Itemize
then 
\begin_inset Formula $\ec f'(x)=27x^{26}$
\end_inset

.
 
\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 That's a 
\emph on
lot
\emph default
 easier than using the precise definition!
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Other forms of 
\begin_inset Formula $x^{n}$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We have shown that the derivative of 
\begin_inset Formula $\ec x^{n}$
\end_inset

 is 
\begin_inset Formula $\ec nx^{n-1}$
\end_inset

 when 
\emph on

\begin_inset Formula $\ec n$
\end_inset

 is a positive integer
\emph default
.
 What about cases where 
\begin_inset Formula $\ec n$
\end_inset

 is a negative integer, or a fraction, or 
\emph on
any
\emph default
 real number?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

The same pattern extends to all 
\emph on
nonzero
\emph default
 real numbers 
\begin_inset Formula $\ec n$
\end_inset

.
 We can say that 
\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f(x)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $f'(x)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
reasoning
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x^{\pi}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\pi x^{\pi-1}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
shortcut
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{1}{x}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-\frac{1}{x^{2}}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{1}{x}=x^{-1}$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sqrt{x}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{1}{2\sqrt{x}}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sqrt{x}=x^{\frac{1}{2}}$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
We 
\emph on
do not
\emph default
 say that the derivative of 
\begin_inset Formula $\ec x^{0}$
\end_inset

 is 
\begin_inset Formula $\ec0x^{-1}$
\end_inset

, because 
\begin_inset Formula $\ec x^{0}=1$
\end_inset

, and the derivative of a constant is 
\begin_inset Formula $\ec0$
\end_inset

.
 So the property of the derivative of 
\begin_inset Formula $\ec x^{n}$
\end_inset

 does not work for every real number.
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The derivative of a polynomial
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Using the properties of derivatives, we can extend this shortcut for 
\emph on
monomials
\emph default
 to a shortcut for 
\emph on
polynomials
\emph default
.
 For example, suppose that 
\begin_inset Formula $\ec f(x)=4x^{2}+3x+2$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Itemize
\begin_inset VSpace -1ex
\end_inset

The shortcut tells us that the derivative of 
\begin_inset Formula $\ec x^{2}$
\end_inset

 is 
\begin_inset Formula $\ec2x$
\end_inset

, and the derivative of 
\begin_inset Formula $\ec x$
\end_inset

 is 
\begin_inset Formula $\ec1$
\end_inset

.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\[
{\dims4{\ec\cancelto{2x}{\left(x^{2}\right)}}+3{\ec\cancelto{1}{\left(x\right)}}+2}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
\begin_inset VSpace -2ex
\end_inset

One property of the derivative tells us that the derivative of the last
 term, 
\begin_inset Formula $\ec2$
\end_inset

, is 
\begin_inset Formula $\ec0$
\end_inset

.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\[
{\dims4\cancelto{2x}{\left(x^{2}\right)}+3\cancelto{1}{\left(x\right)}+{\ec\cancelto{0}{2}}}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
\begin_inset VSpace -2ex
\end_inset

Another property of the derivative tells us that the derivative of 
\begin_inset Formula $\ec4x^{2}$
\end_inset

 is 
\begin_inset Formula $\ec4$
\end_inset

 times the derivative of 
\begin_inset Formula $\ec x^{2}$
\end_inset

, or 
\begin_inset Formula $\ec4\cdot2x=8x$
\end_inset

.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\[
{\dims{\ec\cancelto{8x}{\left(4\cancelto{2x}{\left(x^{2}\right)}\right)}}+3\cancelto{1}{\left(x\right)}+\cancelto{0}{2}}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
\begin_inset VSpace -2ex
\end_inset

Likewise, the derivative of 
\begin_inset Formula $\ec3x$
\end_inset

 is 
\begin_inset Formula $\ec3\cdot1=3$
\end_inset

.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\[
{\dims\cancelto{8x}{\left(4\cancelto{2x}{\left(x^{2}\right)}\right)}+{\ec\cancelto{3}{\left(3\cancelto{1}{\left(x\right)}\right)}}+\cancelto{0}{2}}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Itemize
\begin_inset VSpace -2ex
\end_inset

Another property of the derivative tells us that 
\emph on
the derivative of a sum
\emph default
 is 
\emph on
the sum of the derivatives
\emph default
.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\[
{\ec\cancelto{8x+3+0}{\dims\left(\cancelto{8x}{\left(4\cancelto{2x}{\left(x^{2}\right)}\right)}+\cancelto{3}{\left(3\cancelto{1}{\left(x\right)}\right)}+\cancelto{0}{2}\right)}}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace -2ex
\end_inset


\begin_inset Formula 
\[
{\dims\cancelto{8x+3+0}{\left(\cancelto{8x}{\left(4\cancelto{2x}{\left(x^{2}\right)}\right)}+\cancelto{3}{\left(3\cancelto{1}{\left(x\right)}\right)}+\cancelto{0}{2}\right)}}.
\]

\end_inset

 Thus 
\begin_inset Formula $\ec f'(x)=8x+3$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
We found a shortcut for 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 the derivative of a monomial
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

:
\begin_inset Newline newline
\end_inset

 if 
\begin_inset Formula $\ec f(x)=x^{n}$
\end_inset

 then 
\begin_inset Formula $\ec f'(x)=nx^{n-1}$
\end_inset

.
 
\end_layout

\begin_layout Itemize
This shortcut works for all 
\begin_inset Formula $\ec n\in\mathbb{R}\backslash\{0\}$
\end_inset

.
 
\end_layout

\begin_layout Itemize
To find this shortcut, we used the precise definition of the derivative
 and exploited some patterns that we noticed for 
\begin_inset Formula $\ec n=1,2,3$
\end_inset

.
 
\end_layout

\begin_layout Itemize
We can use properties of the derivative to extend this shortcut to all polynomia
ls.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
