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\begin_layout Title
The derivative of the exponential function
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\begin_layout Subtitle
In this module we determine the derivative of the logarithmic functions
 
\begin_inset Formula $\ec e^{x}$
\end_inset

 and 
\begin_inset Formula $\ec a^{x}$
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, where 
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 is an appropriate constant.
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Introduction
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In this module we determine more efficient ways of computing the derivatives
 of 
\emph on
exponential functions
\emph default
.
 (Recall that an 
\emph on
exponential function
\emph default
 has the form 
\begin_inset Formula $\ec a^{x}$
\end_inset

 where the base 
\begin_inset Formula $\ec a$
\end_inset

 is a positive real number, not equal to 1.
 An important exponential function uses the base 
\begin_inset Formula $\ec e=\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n}\approx2.7183$
\end_inset

.) As with polynomials, we use the precise definition of the derivative.
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Defining the problem
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 How do we evaluate the derivative of an exponential function 
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? 
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SAGE worksheets
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\noindent
You will need a blank SAGE worksheet for this module.
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A difficulty with the precise definition
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name "pg: defining problem"

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 The precise definition of the derivative tells us that
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\[
{\ec f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}}.
\]

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 For an exponential function 
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, this evaluates to
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\[
{\ec f'(x)=\lim_{h\rightarrow0}\frac{a^{x+h}-a^{x}}{h}}.
\]

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 How do we simplify this expression? We would like a shortcut, so as to
 avoid it.
 
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The derivative of 
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We start with a special case, the derivative of 
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.
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:
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\ec f'(x) & {\ec=\lim_{h\rightarrow0}\frac{e^{x+h}-e^{x}}{h}}.
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:
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\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{e^{x+h}-e^{x}}{h}\\
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A property of exponents.
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:
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\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{e^{x}\cdot e^{h}-e^{x}}{h}\\
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Factored a common 
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:
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 we can factor it from the limit.
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The derivative of 
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\[
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 If we could determine the value of the limit, we would be done!
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We cannot substitute 
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 into the expression, because the function is not continuous at 
\begin_inset Formula $\ec h=0$
\end_inset

 (division by zero).
 However, you have learned how to use SAGE to estimate the value of the
 limit.
 Use it now to estimate, 
\emph on
both numerically and graphically
\emph default
, the value of the limit.
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Hint 1:
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 To evaluate it numerically, define the function 
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	filename figures/necessary_limit.png
	lyxscale 20
	scale 30

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The graph appears to confirm our belief that the limit is 1.
\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
We will assume that our intuition is correct.
 Thus
\begin_inset Formula 
\[
\ec f'(x)=e^{x}\cdot1=e^{x}.
\]

\end_inset

 In other words, 
\begin_inset Formula $\ec e^{x}$
\end_inset

 
\emph on
is its own derivative
\emph default
!
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

BIG-TIME FACT
\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

(
\begin_inset Formula $\ec e^{x}$
\end_inset

 is its own derivative)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

 If 
\begin_inset Formula $\ec f(x)=e^{x}$
\end_inset

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

.
 Written another way,
\begin_inset Formula 
\[
{\ec y=e^{x}\qquad\Longrightarrow\qquad y'=y}.
\]

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

We have argued intuitively that 
\begin_inset Formula 
\[
\lim_{h\rightarrow0}\frac{e^{h}-1}{h}=1,
\]

\end_inset

but have not 
\emph on
proved
\emph default
 it.
 It 
\emph on
can
\emph default
 be proved, using the definition
\begin_inset Formula 
\[
{e^{h}=\lim_{x\rightarrow\infty}\left(1+\frac{h}{x}\right)^{x}},
\]

\end_inset

 using an argument similar to (but harder than) the one for the derivative
 of a monomial.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\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 OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

What is the derivative of 
\begin_inset Formula $\ec e^{4x-3}$
\end_inset

?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
We can use the 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 Chain Rule
\emph default
 along with the derivative of 
\begin_inset Formula $\ec e^{x}$
\end_inset

 to answer this.
 Remember that the Chain Rule works with a composition of functions, where
 there is an 
\begin_inset Quotes eld
\end_inset

inside
\begin_inset Quotes erd
\end_inset

 and an 
\begin_inset Quotes eld
\end_inset

outside
\begin_inset Quotes erd
\end_inset

 function.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
Here the 
\begin_inset Quotes eld
\end_inset

inside
\begin_inset Quotes erd
\end_inset

 function is 
\begin_inset Formula $\ec u(x)=4x-3$
\end_inset

, and the 
\begin_inset Quotes eld
\end_inset

outside
\begin_inset Quotes erd
\end_inset

 function is 
\begin_inset Formula $\ec g(u)=e^{u}$
\end_inset

.
\end_layout

\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
From the Chain Rule,
\begin_inset Formula 
\begin{align*}
\ec\frac{d}{dx}e^{4x-3} & {\ec=\frac{dg}{du}\cdot\frac{du}{dx}}.
\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

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
From the Chain Rule,
\begin_inset Formula 
\begin{align*}
\dims\frac{d}{dx}e^{4x-3} & {\dims=\frac{dg}{du}\cdot\frac{du}{dx}}\\
 & {\ec=e^{u}\cdot4}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Substituted for the derivatives.
\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
From the Chain Rule,
\begin_inset Formula 
\begin{align*}
\dims\frac{d}{dx}e^{4x-3} & {\dims=e^{u}\cdot4}\\
 & {\ec=4e^{4x-3}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Substituted for 
\begin_inset Formula $u$
\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
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

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

, for arbitrary 
\begin_inset Formula $a$
\end_inset


\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 determine the derivative of 
\begin_inset Formula $\ec a^{x}$
\end_inset

 for other bases, we use an algebraic technique along with 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 the Chain Rule
\emph default
.
\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
First the algebraic technique.
 Recall that 
\begin_inset Formula $\ec e^{x}$
\end_inset

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

 are inverse functions.
 That is, 
\begin_inset Formula $\ec x=e^{\ln x}$
\end_inset

 for 
\emph on
any
\emph default
 
\begin_inset Formula $\ec x$
\end_inset

.
\end_layout

\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
In that case, we can say that 
\begin_inset Formula 
\begin{align*}
\ec a^{x} & \ec=e^{\ln\left(a^{x}\right)}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Inverse 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
In that case, we can say that 
\begin_inset Formula 
\begin{align*}
\dims a^{x} & \dims=e^{\ln\left(a^{x}\right)}\\
 & \ec=e^{x\ln a}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Property of logarithms.
\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
In that case, we can say that 
\begin_inset Formula 
\begin{align*}
\dims a^{x} & \dims=e^{x\ln a}\\
 & \ec=\left(e^{x}\right)^{\ln a}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Property of exponents.
\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
From the algebraic technique, we have 
\begin_inset Formula $\ec a^{x}=\left(e^{x}\right)^{\ln a}$
\end_inset

.
 Now we'll apply the Chain Rule.
 Here the 
\begin_inset Quotes eld
\end_inset

inside
\begin_inset Quotes erd
\end_inset

 is 
\begin_inset Formula $\ec u(x)=e^{x}$
\end_inset

 and the 
\begin_inset Quotes eld
\end_inset

outside
\begin_inset Quotes erd
\end_inset

 is 
\begin_inset Formula $\ec g(u)=u^{\ln a}$
\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
Thus 
\begin_inset Formula 
\begin{align*}
\ec\frac{d}{dx}\left(e^{x}\right)^{\ln a} & {\ec=\ln a\cdot\left(e^{x}\right)^{\ln a-1}\cdot\left(e^{x}\right)}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

The Chain Rule, and substitution.
\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
Thus 
\begin_inset Formula 
\begin{align*}
\dims\frac{d}{dx}\left(e^{x}\right)^{\ln a} & \dims=\ln a\cdot\left(e^{x}\right)^{\ln a-1}\cdot\left(e^{x}\right)\\
 & {\ec=\ln a\cdot\left(e^{x}\right)^{\ln a\cancelto{0}{-1+1}}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

A property of exponents 
\emph on
that have the same base
\emph default
.
\begin_inset Newline newline
\end_inset

 In this case the base is 
\begin_inset Formula $\ec e^{x}$
\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

<9>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Thus 
\begin_inset Formula 
\begin{align*}
\dims\frac{d}{dx}\left(e^{x}\right)^{\ln a} & \dims=\ln a\cdot\left(e^{x}\right)^{\ln a\cancelto{0}{-1+1}}\\
 & \ec=\ln a\cdot e^{x\ln a}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

A property of exponents.
\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

<10>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Thus 
\begin_inset Formula 
\begin{align*}
\dims\frac{d}{dx}\left(e^{x}\right)^{\ln a} & \dims=\ln a\cdot e^{x\ln a}\\
 & \ec=\ln a\cdot e^{\ln a^{x}}
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

A property of logarithms.
\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

<11>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Thus 
\begin_inset Formula 
\begin{align*}
\dims\frac{d}{dx}\left(e^{x}\right)^{\ln a} & \dims=\ln a\cdot e^{\ln a^{x}}\\
 & {\ec=\ln a\cdot a^{x}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset


\begin_inset Formula $e^{x}$
\end_inset

 and 
\begin_inset Formula $\ln x$
\end_inset

 are inverse functions.
\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

The derivatives of exponential functions
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We have finally found the derivatives we wanted!
\end_layout

\begin_layout Theorem
\noindent
 
\end_layout

\begin_deeper
\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec e^{x}$
\end_inset

 is itself.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec a^{x}$
\end_inset

 is 
\begin_inset Formula $\ec a^{x}\ln a$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Standard
You should commit these to memory.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Examples
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
As with other derivative 
\begin_inset Quotes eld
\end_inset

shortcuts
\begin_inset Quotes erd
\end_inset

, we can use them with the properties of the derivative.
 For example,
\begin_inset Formula 
\begin{align*}
\ec\frac{d}{dx}\left(2e^{x}\right) & \ec=2\frac{d}{dx}e^{x}=2x\\
\ec\frac{d}{dx}\left(xe^{x}\right) & \ec=e^{x}+xe^{x}\\
\ec\frac{d}{dx}\left(\frac{2^{x}}{x^{2}}\right) & \ec=\frac{2^{x}(\ln2)x^{2}-2^{x+1}x}{x^{4}}\\
\ec\frac{d}{dx}\left(e^{x^{2}}\right) & \ec=2x\cdot e^{x^{2}}
\end{align*}

\end_inset

 These illustrate the constant multiple rule, the product rule, the quotient
 rule, and the Chain Rule, respectively.
\end_layout

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

.
 Interestingly, 
\begin_inset Formula $\ec e^{x}$
\end_inset

 is its own derivative.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setlength{
\backslash
shadeboxwidth}{4.5in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Theorem
 If 
\begin_inset Formula $\ec f(x)=e^{x}$
\end_inset

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

.
 Written another way,
\begin_inset Formula 
\[
{\ec y=e^{x}\qquad\Longrightarrow\qquad y'=y}.
\]

\end_inset

 
\end_layout

\end_deeper
\begin_layout Itemize
To find this shortcut, we used the precise definition of the derivative
 and an intuitive estimate of the limit
\begin_inset Formula 
\[
{\ec\lim_{h\rightarrow0}\frac{e^{h}-1}{h}=1}.
\]

\end_inset

 
\end_layout

\begin_layout Itemize
We used the shortcut to find a shortcut for the derivative of 
\begin_inset Formula $\ec a^{x}$
\end_inset

 where the base 
\begin_inset Formula $\ec a$
\end_inset

 is any positive real number, not equal to one.
\end_layout

\begin_deeper
\begin_layout Theorem
The derivative of 
\begin_inset Formula $\ec a^{x}$
\end_inset

 is 
\begin_inset Formula $\ec a^{x}\ln a$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Itemize
To find this second shortcut, we made use of properties of exponents and
 logarithms, along with the Chain Rule.
 
\end_layout

\begin_layout Itemize
These shortcuts can be combined with the properties of derivatives.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
