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\begin_body

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%MODULE: Continuity
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\begin_layout Title
Properties of derivatives
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\begin_layout Subtitle
As with limits, we do not want to spend a lot of time with the precise definitio
n of the derivative, nor with graphical or numerical approximations.
 We prefer to develop properties (or 
\begin_inset Quotes eld
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shortcuts
\begin_inset Quotes erd
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) that allow us to skip the tedious algebra and jump directly into problem
 solving.
 
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%1
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\begin_layout BeginFrame
Defining the problem
\end_layout

\begin_layout Standard
\noindent
We would like properties of the derivative that are similar to properties
 of limits: 
\end_layout

\begin_layout Itemize
the derivative of a constant; 
\end_layout

\begin_layout Itemize
the derivative of a constant multiple; 
\end_layout

\begin_layout Itemize
the derivative of a sum or a difference of functions; 
\end_layout

\begin_layout Itemize
the derivative of a product of functions; and 
\end_layout

\begin_layout Itemize
the derivative of a quotient of functions.
 
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%2
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SAGE worksheets
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}
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\begin_layout Standard
\noindent
You will not need a SAGE worksheet for this module.
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\begin_layout BeginFrame
\begin_inset ERT
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{
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\end_inset

The derivative 
\emph on
as a function
\emph default

\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Until now we have mostly considered the derivative 
\emph on
at a point
\emph default
.
 Recall the definition:
\end_layout

\begin_layout Definition
\begin_inset ERT
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\begin_layout Plain Layout

[
\end_layout

\end_inset

(The derivative at a point)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

 The derivative of a function 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec x=a$
\end_inset

 is 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec{\displaystyle f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}}$
\end_inset

, or equivalently 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec{\displaystyle f'(a)=\lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}}$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
If 
\begin_inset Formula $\ec f$
\end_inset

 is differentiable at every point in its domain, then we can consider the
 derivative 
\emph on
as a function
\emph default
; that is, a formula that describes 
\begin_inset Formula $\ec f'(x)$
\end_inset

 where 
\begin_inset Formula $\ec x$
\end_inset

 is a 
\emph on
variable
\emph default
 instead of a 
\emph on
constant
\emph default
.
 Take the second definition of the derivative at a point, and let the point
 
\begin_inset Formula $\ec a$
\end_inset

 vary.
 What does the definition become?
\end_layout

\begin_layout BeginFrame
The derivative 
\emph on
as a function
\end_layout

\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(The derivative at a point)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

The derivative of a function 
\begin_inset Formula $f$
\end_inset

 at 
\begin_inset Formula $x=a$
\end_inset

 is\SpecialChar \ldots{}
 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula ${\displaystyle f'(a)=\lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}}$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Standard
\noindent
Take the second definition of the derivative at a point, and let the point
 
\begin_inset Formula $\ec a$
\end_inset

 vary.
 What does the definition become?
\end_layout

\begin_layout Definition
\begin_inset ERT
status open

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[
\end_layout

\end_inset

(The derivative as a function)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

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

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

\begin_inset ERT
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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

 
\end_layout

\begin_layout Standard
We simply replaced 
\begin_inset Formula $\ec a$
\end_inset

, which is usually used to represent a constant, with 
\begin_inset Formula $\ec x$
\end_inset

, which is usually used to represent a variable.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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{
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The derivative of a constant
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}
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\begin_deeper
\begin_layout Standard
\noindent
Suppose that 
\begin_inset Formula $\ec f(x)=c$
\end_inset

 where 
\begin_inset Formula $\ec c\in\mathbb{R}$
\end_inset

 is a constant; that is, 
\begin_inset Formula $\ec f(x)$
\end_inset

 has the same value regardless of the value of 
\begin_inset Formula $\ec x$
\end_inset

.
 What is the derivative?
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\begin_deeper
\begin_layout Standard
We can make a guess as to what the derivative 
\emph on
should
\emph default
 be by looking at the graph of a constant, for example 
\begin_inset Formula $\ec f(x)=2$
\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/cplot.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The graph is a horizontal line.
 Any line tangent to this graph should also be horizontal.
 The slope of a horizontal line is zero, so we can venture that
\begin_inset Formula 
\[
{\ec f'(x)=0}.
\]

\end_inset


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\end_layout

\begin_deeper
\begin_layout Standard
The precise definition of the derivative verifies this for us.
\begin_inset Formula 
\begin{align*}
\ec f'(x) & {\ec=\lim_{x\rightarrow0}\frac{f(x+h)-f(x)}{h}}.
\end{align*}

\end_inset


\end_layout

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\end_layout

\begin_deeper
\begin_layout Standard
The precise definition of the derivative verifies this for us.
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{x\rightarrow0}\frac{f(x+h)-f(x)}{h}\\
 & {\ec=\lim_{x\rightarrow0}\frac{c-c}{h}}.
\end{align*}

\end_inset

 
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begin{explanation}
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\begin_inset Formula $f(x)=c$
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\emph on
regardless of
\emph default
 the value of 
\begin_inset Formula $x$
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.
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\end_layout

\begin_deeper
\begin_layout Standard
The precise definition of the derivative verifies this for us.
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{x\rightarrow0}\frac{f(x+h)-f(x)}{h}\\
 & {\ec=\lim_{x\rightarrow0}\frac{\cancelto{0}{c-c}}{h}}.
\end{align*}

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\end_layout

\begin_deeper
\begin_layout Standard
The precise definition of the derivative verifies this for us.
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{x\rightarrow0}\frac{\cancelto{0}{c-c}}{h}\\
 & \ec=\lim_{x\rightarrow0}0\\
 & {\ec=0}.
\end{align*}

\end_inset

 
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begin{explanation}
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The limit of a constant---here, 0---is the constant.
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Properties of derivatives
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}
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\begin_layout Standard
\noindent
We have our first property:
\end_layout

\begin_layout Theorem
\begin_inset ERT
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[
\end_layout

\end_inset

(Properties of derivatives)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

 Let 
\begin_inset Formula $\ec f$
\end_inset

 be a function.
 
\end_layout

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

 is constant, then 
\begin_inset Formula $\ec f'(x)=0$
\end_inset

.
 
\end_layout

\end_deeper
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{
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The derivative of a constant multiple
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}
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\begin_deeper
\begin_layout Standard
\noindent
We move next to the derivative of a constant multiple.
 Rather than make an intuitive argument based on 
\emph on
geometry
\emph default
 here, we can make an intuitive argument based on 
\emph on
properties of limits
\emph default
.
 Let 
\begin_inset Formula $\ec f$
\end_inset

 be a function, 
\begin_inset Formula $\ec c\in\mathbb{R}$
\end_inset

 a constant, and 
\begin_inset Formula $\ec g(x)=cf(x)$
\end_inset

.
 Here 
\begin_inset Formula $\ec g$
\end_inset

 is a 
\emph on
constant multiple
\emph default
 of 
\begin_inset Formula $\ec f$
\end_inset

.
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\begin_deeper
\begin_layout Standard
In an earlier module, we considered the derivatives of 
\begin_inset Formula $\ec\sin x$
\end_inset

 and 
\begin_inset Formula $\hilite4\ec\sin x$
\end_inset

 at 
\begin_inset Formula $\ec x=1$
\end_inset

.
 We found that the first derivative was 
\begin_inset Formula $\ec0.54$
\end_inset

, and the second was 
\begin_inset Formula $\ec2.16=\hilite4\ec\cdot0.54$
\end_inset

.
 This suggests that
\begin_inset Formula 
\[
{\ec\left(g\right)'=cf'(x)}.
\]

\end_inset

(Here 
\begin_inset Formula $\ec f(x)=\sin x$
\end_inset

 and 
\begin_inset Formula $\ec c=4$
\end_inset

.)
\end_layout

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\end_layout

\begin_deeper
\begin_layout Standard
By definition, a derivative is a kind of limit, and from the limit of a
 constant multiple we expect a constant multiple of the limit:
\begin_inset Formula 
\[
{\ec\left(g\right)'=cf'(x)}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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\begin_inset ERT
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<4>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
A close examination of the matter using the precise definition confirms
 this: 
\begin_inset Formula 
\begin{align*}
\ec g'(x) & {\ec=\lim_{h\rightarrow0}\frac{g(x+h)-g(x)}{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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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
A close examination of the matter using the precise definition confirms
 this.
 
\begin_inset Formula 
\begin{align*}
\dims g'(x) & \dims=\lim_{h\rightarrow0}\frac{g(x+h)-g(x)}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{cf(x+h)-cf(x)}{h}}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\end_inset

Substituted 
\begin_inset Formula $g(x)=cf(x)$
\end_inset

.
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<6>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
A close examination of the matter using the precise definition confirms
 this.
 
\begin_inset Formula 
\begin{align*}
\dims g'(x) & \dims=\lim_{h\rightarrow0}\frac{cf(x+h)-cf(x)}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{c\left[f(x+h)-f(x)\right]}{h}}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\end_inset

Factored a common 
\begin_inset Formula $c$
\end_inset

.
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<7>
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\end_layout

\begin_deeper
\begin_layout Standard
\noindent
A close examination of the matter using the precise definition confirms
 this.
 
\begin_inset Formula 
\begin{align*}
\dims g'(x) & \dims=\lim_{h\rightarrow0}\frac{cf(x+h)-cf(x)}{h}\\
 & {\ec=c\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\backslash
begin{explanation}
\end_layout

\end_inset

The constant multiple of a limit.
\begin_inset ERT
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\end_inset


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<8>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
A close examination of the matter using the precise definition confirms
 this.
 
\begin_inset Formula 
\begin{align*}
\dims g'(x) & \dims=c\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\\
 & {\ec=cf'(x)}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\backslash
begin{explanation}
\end_layout

\end_inset

The precise definition of 
\begin_inset Formula $f'(x)$
\end_inset

.
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\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
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{
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Properties of derivatives
\begin_inset ERT
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}
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\end_inset


\end_layout

\begin_layout Standard
\noindent
We have our 
\begin_inset ERT
status collapsed

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{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 second
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 property:
\end_layout

\begin_layout Theorem
\noindent
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

Properties of derivatives
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

 Let 
\begin_inset Formula $\ec f$
\end_inset

 be a function.
 
\end_layout

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

 is constant, then the derivative of 
\begin_inset Formula $\ec f$
\end_inset

 is zero.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec c$
\end_inset

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

 exists,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec cf(x)$
\end_inset

 is 
\begin_inset Formula $\ec cf'(x)$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout BeginFrame
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{
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The derivative of a sum
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<1>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,h$
\end_inset

 be functions such that 
\begin_inset Formula $\ec s=f+g$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec s'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(f+g)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\ec s'(x) & {\ec=\lim_{h\rightarrow0}\frac{s(x+h)-s(x)}{h}}.
\end{align*}

\end_inset


\end_layout

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\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,h$
\end_inset

 be functions such that 
\begin_inset Formula $\ec s=f+g$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec s'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(f+g)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims s'(x) & \dims=\lim_{h\rightarrow0}\frac{s(x+h)-s(x)}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{(f+g)(x+h)-(f+g)(x)}{h}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Definition of 
\begin_inset Formula $s(x)$
\end_inset

.
\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
\noindent
Let 
\begin_inset Formula $\ec f,g,h$
\end_inset

 be functions such that 
\begin_inset Formula $\ec s=f+g$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec s'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(f+g)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims s'(x) & \dims=\lim_{h\rightarrow0}\frac{(f+g)(x+h)-(f+g)(x)}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Definition of addition of functions.
\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
\noindent
Let 
\begin_inset Formula $\ec f,g,h$
\end_inset

 be functions such that 
\begin_inset Formula $\ec s=f+g$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec s'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(f+g)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims s'(x) & \dims=\lim_{h\rightarrow0}\frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}\\
 & {\ec=\lim_{h\rightarrow0}\left(\frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h}\right)}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Group expressions with 
\begin_inset Formula $f$
\end_inset

, 
\begin_inset Formula $g$
\end_inset

.
\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
Let 
\begin_inset Formula $\ec f,g,h$
\end_inset

 be functions such that 
\begin_inset Formula $\ec s=f+g$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec s'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(f+g)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims s'(x) & \dims=\lim_{h\rightarrow0}\left(\frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h}\right)\\
 & {\ec=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}+\lim_{h\rightarrow0}\frac{g(x+h)-g(x)}{h}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

The limit of a sum is the sum of the limits.
\begin_inset Newline newline
\end_inset

 We are assuming that the limits being added exist.
\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

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,h$
\end_inset

 be functions such that 
\begin_inset Formula $\ec s=f+g$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec s'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(f+g)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims s'(x) & \dims=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}+\lim_{h\rightarrow0}\frac{g(x+h)-g(x)}{h}\\
 & \ec=f'(x)+g'(x).
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

The precise definition of the derivative.
\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

Properties of derivatives
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We have our 
\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

 third
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 property: 
\end_layout

\begin_layout Theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

Properties of derivatives
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

Let 
\begin_inset Formula $\ec f$
\end_inset

 be a function.
 
\end_layout

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

 is constant, then the derivative of 
\begin_inset Formula $\ec f$
\end_inset

 is zero.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec c$
\end_inset

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

 exists,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec cf(x)$
\end_inset

 is 
\begin_inset Formula $\ec cf'(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If the derivatives of 
\begin_inset Formula $\ec f(x)$
\end_inset

 and 
\begin_inset Formula $\ec g(x)$
\end_inset

 exist,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec(f\pm g)(x)=f'(x)\pm g'(x)$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The derivative of a product
\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 Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 You might be inclinde to think that the derivative of a product is the
 product of the derivatives; that is,
\begin_inset Formula 
\[
\ec{(f(x)g(x))'=f'(x)g'(x)}.
\]

\end_inset

However, 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 this does not work!
\emph default
 A little pencil grease shows that the derivative of 
\begin_inset Formula $\ec x^{2}$
\end_inset

 is 
\begin_inset Formula $\ec2x$
\end_inset

, so the derivative of 
\begin_inset Formula $\ec x^{2}=x\cdot x$
\end_inset

 cannot be 
\begin_inset Formula $\ec x'\cdot x'=1\cdot1=1$
\end_inset

.
 We have to consider the definition of the derivative.
\end_layout

\end_deeper
\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
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\ec p'(x) & \ec=\lim_{h\rightarrow0}\frac{s(x+h)-s(x)}{h}
\end{align*}

\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
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims p'(x) & \dims=\lim_{h\rightarrow0}\frac{s(x+h)-s(x)}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{(fg)(x+h)-(fg)(x)}{h}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Definition of 
\begin_inset Formula $\ec s(x)$
\end_inset

.
\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
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims p'(x) & \dims=\lim_{h\rightarrow0}\frac{(fg)(x+h)-(fg)(x)}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Definition of the product of two functions.
\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
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims p'(x) & \dims=\lim_{h\rightarrow0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{f(x+h)g(x+h)+{\hilite\left[f(x+h)g(x)-f(x+h)g(x)\right]}-f(x)g(x)}{h}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Added zero: the terms in the middle cancel.
\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

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims p'(x) & \dims=\lim_{h\rightarrow0}\frac{f(x+h)g(x+h)+\left[f(x+h)g(x)-f(x+h)g(x)\right]-f(x)g(x)}{h}\\
 & \ec=\lim_{h\rightarrow0}\frac{f(x+h)\left[g(x+h)-g(x)\right]-g(x)\left[f(x+h)-f(x)\right]}{h}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Factored common terms.
\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

<7>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims p'(x) & \dims=\lim_{h\rightarrow0}\frac{f(x+h)\left[g(x+h)-g(x)\right]-g(x)\left[f(x+h)-f(x)\right]}{h}\\
 & {\ec\overset{?}{=}\lim_{h\rightarrow0}\frac{f(x+h)\left[g(x+h)-g(x)\right]}{h}-\lim_{h\rightarrow0}\frac{g(x)\left[f(x+h)-f(x)\right]}{h}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Properties of limits, if these limits exist.
\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

<8>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\ec p'(x) & {\ec\overset{?}{=}\lim_{h\rightarrow0}\frac{f(x+h)\left[g(x+h)-g(x)\right]}{h}-\lim_{h\rightarrow0}\frac{g(x)\left[f(x+h)-f(x)\right]}{h}}.
\end{align*}

\end_inset

 We need to analyze the limits here.
 In the second case, 
\begin_inset Formula $\ec g(x)$
\end_inset

 is constant with respect to 
\begin_inset Formula $\ec h$
\end_inset

, so 
\begin_inset Formula 
\[
{\ec\lim_{h\rightarrow0}\frac{g(x)\left[f(x+h)-f(x)\right]}{h}=g(x)\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=g(x)f'(x)}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<9>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\ec p'(x) & {\ec\overset{?}{=}\lim_{h\rightarrow0}\frac{f(x+h)\left(g(x+h)-g(x)\right]}{h}-\lim_{h\rightarrow0}\frac{g(x)\left[f(x+h)-f(x)\right]}{h}}.
\end{align*}

\end_inset

 In the first case, we use the properties of the limit of a product: 
\begin_inset Formula 
\[
{\ec\lim_{h\rightarrow0}\frac{f(x+h)\left[g(x+h)-g(x)\right]}{h}=\lim_{h\rightarrow0}f(x+h)\cdot\lim_{h\rightarrow0}\frac{g(x+h)-g(x)}{h}=f(x)\cdot g'(x)}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<10>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f,g,p$
\end_inset

 be functions such that 
\begin_inset Formula $\ec p=fg$
\end_inset

.
 We want to calculate 
\begin_inset Formula $\ec p'(x)$
\end_inset

; that is, the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

.
 From the definition of the derivative, 
\begin_inset Formula 
\begin{align*}
\dims p'(x) & \dims=\lim_{h\rightarrow0}\frac{f(x+h)\left(g(x+h)-g(x)\right]}{h}-\lim_{h\rightarrow0}\frac{g(x)\left[f(x+h)-f(x)\right]}{h}\\
 & {\ec=f(x)g'(x)+f'(x)g(x)}.
\end{align*}

\end_inset

 Notice that we did 
\emph on
not
\emph default
 obtain 
\begin_inset Formula $\ec f'(x)g'(x)$
\end_inset

.
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

Properties of derivatives
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We have our 
\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

 fourth
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 property:
\end_layout

\begin_layout Theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

Properties of derivatives
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

Let 
\begin_inset Formula $\ec f$
\end_inset

 be a function.
 
\end_layout

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

 is constant, then the derivative of 
\begin_inset Formula $\ec f$
\end_inset

 is zero.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec c$
\end_inset

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

 exists,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec cf(x)$
\end_inset

 is 
\begin_inset Formula $\ec cf'(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If the derivatives of 
\begin_inset Formula $\ec f(x)$
\end_inset

 and 
\begin_inset Formula $\ec g(x)$
\end_inset

 exist,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec(f\pm g)(x)$
\end_inset

 is 
\begin_inset Formula $\ec f'(x)\pm g'(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize

\emph on
The product rule
\emph default

\begin_inset Newline newline
\end_inset

 If the derivatives of 
\begin_inset Formula $\ec f(x)$
\end_inset

 and 
\begin_inset Formula $\ec g(x)$
\end_inset

 exist,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

 is 
\begin_inset Formula $\ec f(x)g'(x)+f'(x)g(x)$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The derivative of a quotient
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We will not show all the details of the derivative of a quotient, but the
 argument proceeds in much the same way as the argument for the derivative
 of a product.
 (
\emph on
Try
\emph default
 it!) Instead, we list all the properties on the next page.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Properties of derivatives
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Properties of derivatives)
\begin_inset ERT
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\begin_layout Plain Layout

]
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\end_inset

Let 
\begin_inset Formula $\ec f$
\end_inset

 be a function.
 
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\begin_deeper
\begin_layout Itemize
If 
\begin_inset Formula $\ec f(x)$
\end_inset

 is constant, then the derivative of 
\begin_inset Formula $\ec f$
\end_inset

 is zero.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec c$
\end_inset

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

 exists,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec cf(x)$
\end_inset

 is 
\begin_inset Formula $\ec cf'(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If the derivatives of 
\begin_inset Formula $\ec f(x)$
\end_inset

 and 
\begin_inset Formula $\ec g(x)$
\end_inset

 exist,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec(f\pm g)(x)$
\end_inset

 is 
\begin_inset Formula $\ec f'(x)\pm g'(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize

\emph on
(The product rule)
\emph default

\begin_inset Newline newline
\end_inset

 If the derivatives of 
\begin_inset Formula $\ec f(x)$
\end_inset

 and 
\begin_inset Formula $\ec g(x)$
\end_inset

 exist,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec(fg)(x)$
\end_inset

 is 
\begin_inset Formula $\ec f(x)g'(x)+f'(x)g(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize

\emph on
(The quotient rule)
\emph default

\begin_inset Newline newline
\end_inset

 If the derivatives of 
\begin_inset Formula $\ec f(x)$
\end_inset

 and 
\begin_inset Formula $\ec g(x)$
\end_inset

 exist,
\begin_inset Newline newline
\end_inset

 then the derivative of 
\begin_inset Formula $\ec(f/g)(x)$
\end_inset

 is 
\begin_inset Formula $\ec{\displaystyle \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

Examples
\begin_inset ERT
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}
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\end_inset


\end_layout

\begin_layout Standard
Examples on the following slides use the facts that
\begin_inset Formula 
\[
\mathrm{the\ derivative\ of\ }{\ec x^{2}+1}\mathrm{\ is\ }{\ec2x}\qquad\mathrm{\ and\ the\ derivative\ of\ }{\ec x^{3}-1}\mathrm{\ is\ }{\ec3x^{2}}.
\]

\end_inset

 You should verify these by hand, using the precise definition of the derivative.
\end_layout

\begin_layout BeginFrame
\noindent
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{
\end_layout

\end_inset

Example
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Using the 
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 constant multiple rule
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

, the derivative of 
\begin_inset Formula $\ec4(x^{2}+1)$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\ec4\cdot\left[\frac{d}{dx}(x^{2}+1)\right] & {\ec=4\cdot2x}.
\end{align*}

\end_inset

 (One could simplify this further, but we want you to see the structure.)
\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

You could also verify this by taking the derivative of the product 
\begin_inset Formula $\ec4(x^{2}+1)=4x^{2}+4$
\end_inset

, but at this point it would be laborious to use the precise derivative
 on the product.
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
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\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
Using the 
\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

 product rule
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, the derivative of 
\begin_inset Formula $\ec(x^{2}+1)(x^{3}-1)$
\end_inset

 is
\begin_inset Formula 
\begin{align*}
\ec(x^{2}+1)\left[\frac{d}{dx}(x^{3}-1)\right]+\left[\frac{d}{dx}(x^{2}+1)\right](x^{3}-1) & {\ec=(x^{2}+1)\cdot3x^{2}+2x(x^{3}-1)}.
\end{align*}

\end_inset

 (One could simplify this further, but we want you to see the structure.)
\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

You could also verify this by taking the derivative of the product 
\begin_inset Formula $\ec(x^{2}+1)(x^{3}-1)=x^{5}+x^{3}-x^{2}-1$
\end_inset

 and comparing it to the simplification of the derivative 
\begin_inset Formula $\ec(x^{2}+1)\cdot3x^{2}+2x(x^{3}-1)=5x^{4}+3x^{2}-2x$
\end_inset

, but at this point it would be laborious to use the precise derivative
 on the product.
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

Example
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Using the 
\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

 quotient rule
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, the derivative of 
\begin_inset Formula $\ec(x^{2}+1)/(x^{3}-1)$
\end_inset

 is
\begin_inset Formula 
\[
{\ec\frac{\left[\frac{d}{dx}(x^{2}+1)\right](x^{3}-1)-(x^{2}+1)\left[\frac{d}{dx}(x^{3}-1)\right]}{(x^{3}-1)^{2}}=\frac{2x(x^{3}-1)-(x^{2}+1)\cdot3x^{2}}{(x^{3}-1)^{2}}}.
\]

\end_inset

 (One could simplify this further, but we want you to see the structure.)
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
We determined a definition for the derivative 
\emph on
as a function for arbitrary 
\begin_inset Formula $\ec x$
\end_inset


\emph default
:
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setlength{
\backslash
shadeboxwidth}{4in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(The derivative as a function)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

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

 is differentiable at every point in its domain, then the derivative of
 
\begin_inset Formula $\ec f$
\end_inset

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

\end_inset

 
\end_layout

\end_deeper
\begin_layout Itemize
We identified several properties of the derivative, for 
\end_layout

\begin_deeper
\begin_layout Itemize
a constant and a constant multiple of a function; 
\end_layout

\begin_layout Itemize
a sum or difference of two functions; 
\end_layout

\begin_layout Itemize
a product or quotitent of two functions.
\end_layout

\end_deeper
\begin_layout Itemize
The first three were rather intuitive, based on what we know of limits.
 
\end_layout

\begin_layout Itemize
The last two were counter-intuitive, and required us to play some algebra
 in the precise definition of the derivative.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
