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\centerline{MATH 135 -- Calculus 1}
\centerline{The Derivative Chain Rule}
\centerline{November 4, 2019}
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\noindent
{\it Background}
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Our next major derivative short-cut rule is \emph{one of the most important}.  This 
rule, called the \emph{Chain Rule} allows us to differentiate functions that are 
built up by \emph{composition}.  Here's what it says:  If $g$ is differentiable 
at $x$ and $f$ is differentiable at $g(x)$, then the composition $(f\circ g)(x) = f(g(x))$
is differentiable at $x$ and
$$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x).$$
In words:  The derivative of the composition is \emph{the derivative of the outside
function (i.e. $f'$), with $g(x)$ ``plugged in,'' times the derivative of $g$}.  For example, 
the function 
$$h(x) = \sqrt{x^2 + 4x + 9}$$
is the composition $f(g(x))$ where $f(x) = \sqrt{x}$ and $g(x) = x^2 + 4x + 9$ is ``plugged in.''
The Chain Rule says the derivative of $h$ will be given by computing $f'(x) = \frac{1}{2\sqrt{x}}$ (do you see where that comes from?), plugging $g$ into $f'(x)$, then multiplying by $g'(x)$:
$$h'(x) = \frac{1}{2\sqrt{x^2 + 4x + 9}}\cdot(2x + 4) = \frac{x + 2}{\sqrt{x^2 + 4x + 9}}.$$
Today's class will be devoted to understanding and practicing this rule.  We'll continue
and use this a different way next time. 
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\noindent
{\it Questions}
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\noindent
For each function, identify an $f(x)$ and $g(x)$ such that the given function is the 
composition $f(g(x))$.  Then apply the Chain Rule and compute the derivative:
\begin{enumerate}
\item[(a)] 
$$h(x) = e^{3x + 1}.$$
\item[(b)]
$$h(x) = \frac{1}{(x^4 + 5x^2 + 1)^{3/2}}.$$
\item[(c)]  
$$h(x) = \sin(\cos(x) + x).$$
\item[(d)]
$$h(x) = (\tan(x) + 4x)^3.$$
\item[(e)]  Sometimes we need to use the Chain Rule more than once (if the function
we're looking at is ``several composition layers deep'' like 
$$h(x) = \cos^2(4x^3 + 2) = (\cos(4x^3 + 2))^2.$$
Note that this is $f(g(x))$ with $f(x) = x^2$ and $g(x) = \cos(4x^3 + 2)$.  But 
$g(x)$ is also a composition, so you'll need to use the Chain Rule again to find $g'(x)$.
With these hints, find $h'(x)$.  
\end{enumerate}
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