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\title{Chapter 3, Differentiation}

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\section*{secant lines, definition of derivative}

(two notations: $x \to a$ and $h \to 0$)

\section*{differentiation rules: vector addition,
chain rule, multiplication, inverse}

$\frac{d}{dx}x^n=nx^{n-1}$ -- this is true for $n$ any real number,
(actually: for $n$ an integer, this works for complex numbers;
 for $n$ a rational number it's a bit tricky to define rational powers
 for real numbers, it's \emph{really} bad...)
so thinking about function $\bR \to \bR$, that works for any real number

\section*{derivative of several variables}

\section*{multiple derivatives, functions of several variables}

(note that sum rule and product rule come from derivative of
sum and product function plus chain rule...)

\section*{derivative as rate of change, e.g., velocity}
velocity, acceleration (velocity in several dimensions)
speed (jerk, pulse)

\section*{derivatives of trig functions}
(get derivative of sin and cosine from circle, rest from that...)

\section*{Partial Derivatives; Differentiating Functions in Several Variables}
...
the amount a function in several variables changes as you vary
$x$ and $y$ is by $df$, given by...

$df=\frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$

(implicitly there's linear algebra going on...)

\section*{Implicit Differentiation}
Given a function $f\from \bR^2 \to \bR$, we can graph $f$ as a surface, where
$z=f(x,y)$ is the ``height'' of the graph at $(x,y)$.
Given a height $c\in \bR$, we can look at
$f^{-1}(c)=\set{(x,y) : f(x,y)=c}$, which we call a \Def{level set},
as you cut (level) at height $c$.

Recall that if you vary a differentiable function by $(\Delta x,\Delta y)$,
then the value of the function varies by approximately
$$\Delta f \approx \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y$$
Infintesimally, this is exactly the total derivative:
$$df=\frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$$
So suppose you want to stay on the level set $f(x,y)=c$, i.e., you want
$df = 0$ -- this is exactly the derivative of the equation $f=c$,
so the equation of the tangent line \emph{of the level set} at $(x_0,y_0)$
is
$$0=\frac{\partial f}{\partial x}(x_0,y_0) dx
  + \frac{\partial f}{\partial y}(x_0,y_0) dy$$
This is called ``implicit differentiation'', as you can now solve for
$\frac{dy}{dx}$ or $\frac{dx}{dy}$ to find the slope of a curve defined by a
relation (``implicitly'') instead of as a function.

In higher dimensions, you can express the tangent plane as being orthogonal to
grad.

\section*{Related Rates}
these are just implicit differentiation story problems

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