picture yourself as an early calculus
student about to begin your first course
the months ahead of you hold within them
a lot of hard work some neat examples
some not so neat examples beautiful
connections to physics not so beautiful
piles of formulas to memorize plenty of
moments of getting stuck and banging
your head into a wall a few nice aha
moments sprinkled in as well and some
genuinely lovely graphical intuition to
help guide you through it all but if the
course ahead of you is anything like my
first introduction to calculus or any of
the first courses that I've seen in the
years since there's one topic that you
will not see but which I believe stands
to greatly accelerate your learning you
see almost all of the visual intuitions
from that first year are based on graphs
the derivative is the slope of a graph
the integral is a certain area under
that graph but as you generalize
calculus beyond functions whose inputs
and outputs are simply numbers it's not
always possible to graph the function
that you're analyzing
there's all sorts of different ways that
you'd be visualizing these things so if
all your intuitions for the fundamental
ideas like derivatives are rooted too
rigidly in graphs it can make for a very
tall and largely unnecessary conceptual
hurdle between you and the more
quote/unquote advanced topics like
multivariable calculus and complex
analysis differential geometry now what
I want to share with you is a way to
think about derivatives which I'll refer
to as the transformational view that
generalizes more seamlessly into some of
those more general context where
calculus comes up and then we'll use
this alternate view to analyze a certain
fun puzzle about repeated fractions but
first off I just want to make sure that
we're all on the same page about what
the standard visual is if you were to
graph a function which simply takes real
numbers as inputs and outputs one of the
first things you learn in a calculus
course is that the derivative gives you
the slope of this graph where what we
mean by that is that the derivative of
the function is a new function which for
every input X returns that slope now I'd
encourage you not to think of this
derivative as slope idea as being the
definition of a derivative instead think
of it as being more fundamentally about
how sensitive the function is to tiny
little nutters around the input and the
slope is just one way to think about
that since
divot II relevant only to this
particular way of viewing functions I
have not just another video but a full
series on this topic if it's something
you want to learn more about now the
basic idea behind the alternate visual
for the derivative is to think of this
function as mapping all of the input
points on the number line to their
corresponding outputs on a different
number line in this context what the
derivative gives you is a measure of how
much the input space gets stretched or
squished in various regions that is if
you were to zoom in around a specific
input and take a look at some evenly
spaced points around it the derivative
of the function of that input is going
to tell you how spread out or contracted
those points become after the mapping
here a specific example helps take the
function x squared it maps 1 to 1 and 2
to 4 3 to 9 and so on and you could also
see how it acts on all of the points in
between and if you were to zoom in on a
little cluster of points around the
input 1 and then see where they land
around the relevant output which for
this function also happens to be 1 you'd
notice that they tend to get stretched
out in fact it roughly looks like
stretching out by a factor of 2 and the
closer you zoom in the more this local
behavior looks just like multiplying by
a factor of 2 this is what it means for
the derivative of x squared at the input
x equals 1 to be 2 it's what that fact
looks like in the context of
transformations if you looked at a
neighborhood of points around the input
3 they would get roughly stretched out
by a factor of 6 this is what it means
for the derivative of this function at
the input 3 to equal 6 around the input
1/4 a small region actually tends to get
contracted
specifically by a factor of 1/2 and
that's what it looks like for a
derivative to be smaller than 1 now the
input 0 is interesting zooming in by a
factor of 10 it doesn't really look like
a constant stretching or squishing for
one thing all of the outputs end up on
the right positive side of things
and as you zoom in closer and closer by
100x or by 1000 X it looks more and more
like a small neighborhood of points
around zero just gets collapsed into
zero itself and this is what it looks
like for the derivative to be zero the
local behavior looks more and more like
multiplying the whole number line by
zero it doesn't have to completely
collapse everything to a point at a
particular zoom level instead it's a
matter of what the limiting behavior is
as you zoom in closer and closer it's
also instructive to take a look at the
negative inputs here things start to
feel a little cramped since they collide
with where all the positive input values
go and this is one of the downsides of
thinking of functions as transformations
but for derivatives we only really care
about the local behavior anyway what
happens in a small range around a given
input here notice that the inputs in a
little neighborhood around say negative
two they don't just get stretched out
they also get flipped around
specifically the action on such a
neighborhood looks more and more like
multiplying by negative for the closer
you zoom in this is what it looks like
for the derivative of a function to be
negative and I think you get the point
this is all well and good but let's see
how this is actually useful in solving a
problem a friend of mine recently asked
me a pretty fun question about the
infinite fraction one plus one divided
by one plus one divided by one plus one
divided by one on and on and on and on
and clearly you watch math videos online
so maybe you've seen this before but my
friend's question actually cuts to
something that you might not have
thought about before relevant to the
view of derivatives that we're looking
at here the typical way that you might
evaluate an expression like this is to
set it equal to X and then notice that
there's a copy of the full fraction
inside itself so you can replace that
copy with another X and then just solve
for X that is what you want is to find a
fixed point of the function 1 plus 1
divided by X
but here's the thing there are actually
two solutions for X two special numbers
were one plus one divided by that number
gives you back the same thing one is the
golden ratio fee around 1.618
and the other is negative 0.618 which
happens to be negative 1 divided by fee
I like to call this other number fees
little brother since just about any
property that fee has this number also
has and this raises the question would
it be valid to say that that infinite
fraction that we saw is somehow also
equal to fees little brother negative
0.618
maybe you initially say obviously not
everything on the left hand side is
positive so how could it possibly equal
a negative number
well first we should be clear about what
we actually mean by an expression like
this one way that you could think about
it then it's not the only way there's
freedom for choice here is to imagine
starting with some constant like 1 and
then repeatedly applying the function 1
plus 1 divided by X and then asking what
is this approach as you keep going I
mean certainly symbolically what you get
looks more and more like our infinite
fraction so maybe if you wanted to equal
a number you should ask what this series
of numbers approaches and if that's your
view of things maybe you start off with
a negative number so it's not so crazy
for the whole expression to end up
negative after all if you start with
negative 1 divided by fie then applying
this function 1 plus 1 over X you get
back the same number negative 1 divided
by fee so no matter how many times you
apply it you're staying fixed at this
value
but even then there is one reason that
you should probably view fee as the
favorite brother in this pair here try
this pull up a calculator of some kind
then start with any random number and
then plug it into this function 1 plus 1
divided by X and then plug that number
into 1 plus 1 over X and then again and
again and again and again and again no
matter what constant you start with you
eventually end up at 1.618
even if you start with a negative number
even one that's really really close to
fees little brother
eventually it shiz away from that value
and jumps back over to fee
so what's going on here why is one of
these fixed points favored above the
other one maybe you can already see how
the transformational understanding of
derivatives is going to be helpful for
understanding this set up but for the
sake of having a point of contrast I
want to show you how a problem like this
is often taught using graphs if you were
to plug in some random input to this
function the y-value tells you the
corresponding output right so to think
about plugging that output back into the
function you might first move
horizontally until you hit the line y
equals x and that's going to give you a
position where the x value corresponds
to your previous Y value right so then
from there you can move vertically to
see what output this new x value has and
then you repeat you move horizontally to
the line y equals x to find a point
whose x value is the same as the output
that you just got and then you move
vertically to apply the function again
now personally I think this is kind of
an awkward way to think about repeatedly
applying a function don't you I mean it
makes sense but you can't have to pause
and think about it to remember which way
to draw the lines and you can if you
want think through what conditions make
this spiderweb process narrow in on a
fixed point versus propagating away from
it and in fact go ahead pause right now
and try to think it through as an
exercise it has to do with slopes
or if you want to skip the exercise for
something that I think gives a much more
satisfying understanding think about how
this function acts as a transformation
so I'm gonna go ahead and start here by
drawing a whole bunch of arrows to
indicate where the various sample the
input points will go and side note don't
you think this gives a really neat
emergent pattern I wasn't expecting this
but it was cool to see it pop up when
animating I guess the action of 1
divided by X gives this nice emergent
circle and then we're just shifting
things over by 1 anyway I want you to
think about what it means to repeatedly
apply some function like 1 plus 1 over X
in this context well after letting it
map all of the inputs to the outputs you
could consider those as the new inputs
and then just apply the same process
again and then again and do it however
many times you want notice in animating
this with a few dots representing the
sample points it doesn't take many
iterations at all before all of those
dots kind of clump in around 1.618
now remember we know that 1.618 and its
little brother negative 0.618 on and on
stay fixed in place during each
iteration of this process but zoom in on
a neighborhood around fee during the map
points in that region
get contracted around fee meaning that
the function 1 plus 1 over X has a
derivative with a magnitude that's less
than 1 at this input in fact this
derivative works out to be around
negative 0.38 so what that means is that
each repeated application scrunches the
neighborhood around this number smaller
and smaller like a gravitational pull
towards feet so now tell me what you
think happens in the neighborhood of
these little brother
over there the derivative actually has a
magnitude larger than one so points near
the fixed point are repelled away from
it and when you work it out you can see
that they get stretched by more than a
factor of two in each iteration they
also get flipped around because the
derivative is negative here but the
salient fact for the sake of stability
is just the magnitude mathematicians
would call this right value a stable
fixed point and the left one is an
unstable fixed point something is
considered stable if when you perturb it
just a little bit it tends to come back
towards where it started rather than
going away from it so what we're seeing
is a very useful little fact that the
stability of a fixed point is determined
by whether or not the magnitude of its
derivative is bigger or smaller than one
and it explains why fee always shows up
in the numerical play where you're just
hitting enter on your calculator over
and over but fees little brother never
does now as to whether or not you want
to consider fees little brother a valid
value of the infinite fraction well
that's really up to you everything we
just showed suggests that if you think
of this expression as representing a
limiting process then because every
possible seed value other than fees
little brother gives you a series
converting to fee it does feel kind of
silly to put them on equal footing with
each other but maybe you don't think of
it as a limit maybe the kind of math
you're doing lends itself to treating
this as a purely algebraic object like
the solutions of a polynomial which
simply has multiple values anyway that's
beside the point and my point here is
not that viewing derivatives as this
change in density is somehow better than
the graphical intuition on the whole in
fact picturing an entire function this
way can be kind of clunky and
impractical as compared to graphs my
point is that it deserves more of a
mention in most of the introductory
calculus courses because it can help
make a student's understanding of the
derivative a little bit more flexible
like I mentioned the real reason that
I'd recommend you carry this perspective
with you as you learn new topics is not
so much for what it does with your
understanding of single variable
calculus it's for what comes after there
are many topics typically taught in a
college math department which actually
put this lately don't exactly have a
reputation of being super accessible
so in the next video I'm gonna show you
how a few ideas from these subjects with
fancy sounding names like holomorphic
functions and the Jacobian determinant
are really just extensions of the idea
shown here
they really are some beautiful ideas
which I think can be appreciated from a
really wide range of mathematical
backgrounds and they're relevant to a
surprising number of seemingly unrelated
ideas so stay tuned for that
now for the final animation I just want
to show you a little more of that
time-dependent vector field I flashed
earlier but first let's look at some of
the principles of learning from this
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curiosity I love the word choice here
it's not just that you should be curious
in one moment it means creating a
context where that curiosity is
constantly growing just look at the
infinite fraction example here it would
be one thing if you were curious about
why the numbers bounce around the way
that they do but hopefully the
conclusion is not just to understand
this one example I would want you to
start looking at all sorts of other
infinite expressions and wonder if
there's some fixed point phenomenon in
them or wonder where else this view of
derivatives can be conceptually helpful
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[Music]
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