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%MODULE: Continuity
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\begin_layout Title
The Chain Rule
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\begin_layout Subtitle
In this module we turn to another property of derivatives, the 
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{
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hilite
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 Chain Rule
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}
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.
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%1
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Defining the problem
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begin{question}
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How can we compute the derivatives of the following functions 
\emph on
quickly
\emph default
?
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\begin_layout Itemize
\begin_inset Formula $\ec\sin(x^{2}-1)$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\left(x^{2}-x+1\right)^{100}$
\end_inset

 
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SAGE worksheets
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\noindent
You do not need a SAGE worksheet for this module.
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{
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The Chain Rule
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\begin_layout Standard
\noindent
We start by stating 
\emph on

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 the Chain Rule
\emph default
:
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\begin_layout Theorem
\begin_inset ERT
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\begin_layout Plain Layout

[(The Chain Rule)]
\end_layout

\end_inset


\end_layout

\begin_layout Theorem
\noindent
Let 
\begin_inset Formula $\ec g$
\end_inset

 and 
\begin_inset Formula $\ec u$
\end_inset

 be functions such that 
\begin_inset Formula $\ec u$
\end_inset

 is differentiable on its domain and 
\begin_inset Formula $\ec g$
\end_inset

 is differentiable on the range of 
\begin_inset Formula $\ec u$
\end_inset

.
 If 
\begin_inset Formula $\ec f=g\circ u$
\end_inset

 then 
\begin_inset Formula 
\[
{\ec f'(x)=g'(u)\cdot u'(x)}.
\]

\end_inset

 Alternatively, in Leibniz notation
\begin_inset Formula 
\[
{\ec\frac{df}{dx}=\frac{dg}{du}\cdot\frac{du}{dx}}.
\]

\end_inset


\end_layout

\begin_layout Standard
\noindent
The Leibniz notation does 
\emph on
not
\emph default
 prove the theorem, although it suggests it.
 We will not prove this theorem (your text or your instructor can provide
 a proof) but we will review quite a few examples.
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{
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A note on the Chain Rule
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}
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\end_layout

\begin_layout Standard
\noindent
Look back at the Chain Rule and observe the structure of 
\begin_inset Formula $\ec f$
\end_inset

: it is a 
\emph on
composition of functions
\emph default
.
 The Chain Rule's strength is that it helps us compute derivatives of functions
 with this structure.
 The examples given at the beginning of the module are of this kind:
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\begin_layout Standard
\noindent

\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\sin(x^{2}-1)=(g\circ u)(x)$
\end_inset

 where
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec g(u)=\sin u$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u(x)=x^{2}-1$
\end_inset

.
 
\end_layout

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\begin_layout Pause

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\begin_layout Itemize
\begin_inset Formula $\ec\left(x^{2}-x+1\right)^{100}=(g\circ u)(x)$
\end_inset

 where
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec g(u)=u^{100}$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u(x)=x^{2}-x+1$
\end_inset

.
 
\end_layout

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\begin_layout Standard
\noindent
Compositions of functions often look as if they have 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

; here 
\begin_inset Formula $\ec u$
\end_inset

 is the 
\begin_inset Quotes eld
\end_inset

inside
\begin_inset Quotes erd
\end_inset

 function and 
\begin_inset Formula $\ec g$
\end_inset

 is the 
\begin_inset Quotes eld
\end_inset

outside
\begin_inset Quotes erd
\end_inset

 function.
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First Example
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\noindent
In the 
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{
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\end_inset


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\backslash
hilite
\end_layout

\end_inset

 first
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 example, we have 
\begin_inset Formula $\ec f(x)=\sin(x^{2}-1)=(g\circ u)(x)$
\end_inset

 where
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec g(u)=\sin u$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u(x)=x^{2}-1$
\end_inset

.
 
\end_layout

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<2>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Application of the Chain Rule gives
\begin_inset Formula 
\begin{align*}
\ec f'(x) & {\ec=g'(u)\cdot u'(x)}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
begin{explanation}
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\end_inset

Definition of the Chain Rule.
\begin_inset ERT
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end{explanation}
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\end_inset


\end_layout

\end_deeper
\begin_layout Pause

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\begin_inset ERT
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<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Application of the Chain Rule gives
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=g'(u)\cdot u'(x)\\
 & {\ec=\cos u\cdot(2x)}.
\end{align*}

\end_inset

 
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begin{explanation}
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\end_inset

Substituted the derivatives.
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end{explanation}
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\end_inset


\end_layout

\end_deeper
\begin_layout Pause

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<4>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Application of the Chain Rule gives
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\cos u\cdot(2x)\\
 & {\ec=2x\cos\left(x^{2}-1\right)}.
\end{align*}

\end_inset

 
\begin_inset ERT
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begin{explanation}
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\end_inset

Substituted for 
\begin_inset Formula $u$
\end_inset

.
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\end_deeper
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{
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\end_inset

Second Example
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\begin_deeper
\begin_layout Standard
\noindent
In the 
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{
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\end_inset


\begin_inset ERT
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\backslash
hilite
\end_layout

\end_inset

 second
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 example, we have 
\begin_inset Formula $\ec f(x)=\left(x^{2}-x+1\right)^{100}=(g\circ u)(x)$
\end_inset

 where
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec g(u)=u^{100}$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u(x)=x^{2}-x+1$
\end_inset

.
\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
Application of the Chain Rule gives 
\begin_inset Formula 
\begin{align*}
\ec f'(x) & {\ec=g'(u)\cdot u'(x)}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Definition of the Chain Rule.
\begin_inset ERT
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\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
Application of the Chain Rule gives 
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=g'(u)\cdot u'(x)\\
 & {\ec=100u^{99}\cdot(2x)}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\backslash
begin{explanation}
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\end_inset

Substituted the derivatives.
\begin_inset ERT
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end{explanation}
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\end_deeper
\begin_layout Standard
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\end_inset


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\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
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\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Application of the Chain Rule gives 
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=100u^{99}\cdot(2x)\\
 & {\ec=200\left(x^{2}-x+1\right)^{99}}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\backslash
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\end_layout

\end_inset

Substituted for 
\begin_inset Formula $u$
\end_inset

.
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\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
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\begin_layout BeginFrame
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{
\end_layout

\end_inset

Third Example
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let's try a third example.
 Let 
\begin_inset Formula $\ec f(x)=\tan\left((x-3)^{3}+(x-3)^{2}-1\right)$
\end_inset

.
 Start by identifying 
\begin_inset Formula $\ec f$
\end_inset

 and 
\begin_inset Formula $\ec u$
\end_inset

 in this function.
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<2-7>
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\begin_deeper
\begin_layout Standard
We have 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec g(u)=\tan u$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u(x)=(x^{2}-3)^{3}+(x^{2}-3)^{2}-1$
\end_inset

.
\end_layout

\end_deeper
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\end_layout

\end_inset


\end_layout

\begin_layout Only
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<2>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
(It is possible to obtain a different answer.
 Check with your instructor if you think you have a different answer that
 is correct.)
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Notice that 
\begin_inset Formula $\ec u(x)$
\end_inset

 is also a composition of functions! Before we can compute 
\begin_inset Formula $\ec f'(x)$
\end_inset

 we will have to compute 
\begin_inset Formula $\ec u'(x)$
\end_inset

.
\end_layout

\end_deeper
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<4-7>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Put 
\begin_inset Formula $\ec u(x)=(g_{1}\circ u_{1})(x)$
\end_inset

 where
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec g_{1}(u_{1})=u_{1}^{3}+u_{1}^{2}-1$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u_{1}(x)=x^{2}-3$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
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<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Then
\begin_inset Formula 
\begin{align*}
\ec u'(x) & {\ec=g_{1}'(u_{1})\cdot u_{1}'(x)}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\backslash
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\end_layout

\end_inset

Definition of the Chain Rule.
\begin_inset ERT
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\backslash
end{explanation}
\end_layout

\end_inset

 
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\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Then
\begin_inset Formula 
\begin{align*}
\dims u'(x) & {\dims=g_{1}'(u_{1})\cdot u_{1}'(x)}\\
 & {\ec=(3u_{1}^{2}+2u_{1})\cdot2x}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\end_layout

\end_inset

Substituted the derivatives.
\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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\begin_layout Plain Layout

<7>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Then
\begin_inset Formula 
\begin{align*}
\dims u'(x) & {\dims=(3u_{1}^{2}+2u_{1})\cdot2x}\\
 & {\ec=2x\left[3(x^{2}-3)^{2}+2(x^{2}-3)\right]}.
\end{align*}

\end_inset

 
\begin_inset ERT
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\end_layout

\end_inset

Substituted for 
\begin_inset Formula $u_{1}$
\end_inset

.
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\backslash
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\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

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<8->
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We have 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec g(u)=\tan u$
\end_inset

,
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u(x)=(x^{2}-3)^{3}+(x^{2}-3)^{2}-1$
\end_inset

, and
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec u'(x)=2x\left[3(x^{2}-3)^{2}+2(x^{2}-3)\right]$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Standard
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\end_layout

\end_inset


\end_layout

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<8>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We finally put them together:
\begin_inset Formula 
\begin{align*}
\ec f'(x) & {\ec=g'(u)\cdot u'(x)}.\\
 & \phantom{=\sec^{2}\left[(x^{2}-3)^{3}+(x^{2}-3)^{2}-1\right]\cdot2x\left[3(x^{2}-3)^{2}+2(x^{2}-3)\right]}
\end{align*}

\end_inset

 
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The Chain Rule.
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<9>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We finally put them together:
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=g'(u)\cdot u'(x)\\
 & {\ec=\sec^{2}u\cdot2x\left[3(x^{2}-3)^{2}+2(x^{2}-3)\right]}.\\
 & \phantom{=\sec^{2}\left[(x^{2}-3)^{3}+(x^{2}-3)^{2}-1\right]\cdot2x\left[3(x^{2}-3)^{2}+2(x^{2}-3)\right]}
\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 $g'$
\end_inset

 and 
\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
\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
We finally put them together:
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\sec^{2}u\cdot2x\left[3(x^{2}-3)^{2}+2(x^{2}-3)\right]\\
 & {\ec=\sec^{2}\left[(x^{2}-3)^{3}+(x^{2}-3)^{2}-1\right]\cdot2x\left[3(x^{2}-3)^{2}+2(x^{2}-3)\right]}.
\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
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Numerical 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
\noindent
In the fourth example, the formulas for 
\begin_inset Formula $\ec f$
\end_inset

, 
\begin_inset Formula $\ec g$
\end_inset

, and 
\begin_inset Formula $\ec u$
\end_inset

 are not known, but the following values 
\emph on
are
\emph default
 known.
 
\end_layout

\begin_layout Standard
\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="6" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<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" topline="true" leftline="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" rightline="true" usebox="none">
\begin_inset Text

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

 
\end_layout

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

\begin_layout Plain Layout
-2 
\end_layout

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

\begin_layout Plain Layout
-2 
\end_layout

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

\begin_layout Plain Layout
10 
\end_layout

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

\begin_layout Plain Layout
5 
\end_layout

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

\begin_layout Plain Layout
-10 
\end_layout

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

\begin_layout Plain Layout
-1 
\end_layout

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

\begin_layout Plain Layout
5 
\end_layout

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

\begin_layout Plain Layout
2 
\end_layout

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

\begin_layout Plain Layout
-3 
\end_layout

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

\begin_layout Plain Layout
-5 
\end_layout

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

\begin_layout Plain Layout
0 
\end_layout

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

\begin_layout Plain Layout
7 
\end_layout

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

\begin_layout Plain Layout
0 
\end_layout

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

\begin_layout Plain Layout
0 
\end_layout

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

\begin_layout Plain Layout
-4 
\end_layout

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

\begin_layout Plain Layout
1 
\end_layout

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

\begin_layout Plain Layout
6 
\end_layout

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

\begin_layout Plain Layout
-3 
\end_layout

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

\begin_layout Plain Layout
-2 
\end_layout

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

\begin_layout Plain Layout
0 
\end_layout

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

\begin_layout Plain Layout
2 
\end_layout

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

\begin_layout Plain Layout
0 
\end_layout

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

\begin_layout Plain Layout
-2 
\end_layout

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

\begin_layout Plain Layout
-1 
\end_layout

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

\begin_layout Plain Layout
3 
\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
If 
\begin_inset Formula $\ec f(x)=(g\circ u)(x)$
\end_inset

, what is 
\begin_inset Formula $\ec f'(2)$
\end_inset

?
\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
Again, we use the Chain Rule.
 We have
\begin_inset Formula 
\begin{align*}
\ec f'(x) & {\ec=g'(u)\cdot u'(x)}.
\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
We want values of 
\begin_inset Formula $\ec f'(x)$
\end_inset

 when 
\begin_inset Formula $\ec x=2$
\end_inset

, or
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=g'(u)\cdot u'(x)\\
\ec f'(2) & {\ec=g'(u(2))\cdot u'(2)}.
\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

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We can determine those values from the table:
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=g'(u)\cdot u'(x)\\
\dims f'(2) & \dims=g'(u(2))\cdot u'(2)\\
 & {\ec=g'(-1)\cdot3=2\cdot3=6}.
\end{align*}

\end_inset

So 
\begin_inset Formula $\ec f'(2)=6$
\end_inset

.
\end_layout

\end_deeper
\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
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

 Chain Rule
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 is 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

 functions derived by composition
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

:
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setlength{
\backslash
shadeboxwidth}{3.5in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

[(The Chain Rule)]
\end_layout

\end_inset

Let 
\begin_inset Formula $\ec g$
\end_inset

 and 
\begin_inset Formula $\ec u$
\end_inset

 be functions such that 
\begin_inset Formula $\ec u$
\end_inset

 is differentiable on its domain and 
\begin_inset Formula $\ec g$
\end_inset

 is differentiable on the range of 
\begin_inset Formula $\ec u$
\end_inset

.
 If 
\begin_inset Formula $\ec f=g\circ u$
\end_inset

 then 
\begin_inset Formula 
\[
{\ec f'(x)=g'(u)\cdot u'(x)}.
\]

\end_inset

 Alternatively, in Leibniz notation
\begin_inset Formula 
\[
{\ec\frac{df}{dx}=\frac{dg}{du}\cdot\frac{du}{dx}}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
To use the Chain Rule, we must identify an 
\begin_inset Quotes eld
\end_inset


\emph on
inside
\emph default

\begin_inset Quotes erd
\end_inset

 function and an 
\begin_inset Quotes eld
\end_inset


\emph on
outside
\emph default

\begin_inset Quotes erd
\end_inset

 function.
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
