Hi, guys! I'm Nancy. And I'm going to
show you what the derivative is, the
definition of the derivative with the
limits and how to find the derivative
with the definition. And I'm going to show
you all the important parts step by step
so you won't miss anything. And I know
that for some of you that may sound like a
threat, but don't worry it's not that
bad. I promise. Let me show you.
So what is the derivative? Long story short it's
a function that tells you the slope of the
line tan gent to the curve at any point. It
gives you the rate of change at any
instant, the instantaneous rate of change
at any point. And we're going to figure
out the proper definition of the
derivative with the limit. But if you
just can't wait to use it to calculate the
derivative, you can skip ahead and jump to
the time in the description. Or you can go
down this rabbit hole with us. But say you
have a random curve. The derivative
tells you the slope at any
point on your curve, the steepness or rise
over run - the slope. But really what it's
telling you is the slope of the line
tangent to the curve. So a tangent line is a
line that touches at only one point on the
curve. It skims the  surface, grazes the
curve and touches at only one point. And
every point on your graph has a different
little tangent line. So if the
derivative's supposed to tell you the
slope at each little tangent line, how can
we know that slope if we only know one
point for sure on each tangent line? And
you know we need two points to find the
slope. Very good question. We don't know
how to do that yet. I mean if we had
instead a straight-line graph a linear
function like that, we would know the 
slope everywhere because you could pick
any two points there, use the slope
formula, get a number for slope and it
would be the same everywhere that slope.
You'd be done. But life is not that simple
usually. Our graph is curved and the slope
is changing everywhere. The tangent line
is different everywhere. So how can we
know the slope everywhere if it's always
changing and we only know one point for
sure on each tangent line? Like I said, we
don't know how to do that yet. We are not
that sophisticated yet. But we can figure
it out from things that we do know and we
can make the actual definition or formula
for the derivative. OK, so for the derivative
if we want to know the slope of this
tangent line at any point call  it x we
can start with what we do know the good
old slope of the straight line between two
points. Pick a second point somewhere on
the curve and draw a straight line
through. And we know how to find the slope
of a straight line with the old-fashioned
slope formula, which is a throwback to
middle school probably. So finding the
slope between two points has come back to
haunt you. But you have to know your
points. So let's label them. If this
second point is some distance away
horizontally, let's call that distance h,
then this point has an x coordinate that
is h more than x or x + h. We want the
actual points the x, y pairs. So for this
point, for the x, y pair, the x coordinate
will be x. And the y coordinate we'll
call f(x) which just stands for the y
value that the function will give us.
So that's that point. And for this other
point, for the x, y pair, the x coordinate
will be not x but x + h. And the y
coordinate will be not f(x), but f(x) + h.
And just humor me. This is all going
somewhere. We have two points, which is
great, because we can use those two points
in the slope formula that you know and love.
OK, so here's the slope formula
it's the rise over run or vertical change
over the horizontal change. It's y2 - y1
all over the x2 -  x1.
So let's use that for our two points.
OK, so this is the slope we
found for the secant line. I did some
simplifying and you could cancel 'x's in
the bottom but this is the slope and it
ended up being the difference in the 'y's
over the horizontal difference, the h.
So it's the slope of the secant line. And
it's also known as the difference quotient
if you hear that because it's a quotient
of differences. Very original name.
This is the slope of the secant line. And we
are halfway to the derivative so bear with me.
So we are almost to the derivative.
And we would be there. We would be done if
we had a straight-line graph. A linear
function, then this slope we found would
be right everywhere. It would be enough.
But we don't have that we have a curved
graph. And the straight line secant slope
that we found it's actually an okay
approximation for the slope at that point
x. It's a rough estimate. And that's a
big part of
calculus is estimating something nonlinear
with something linear. So it's a rough
estimate, the slope we found. And it's not
great but it's decent. But let's not
settle for decent. We don't want an
approximation of the slope here. We want
the exact slope, the real slope there and
not some wonky approximation between it
and some point nearby. No, we can make it
exact if we close in on x by narrowing h
to zero and picking a right point that is
closer and closer to the left so that the
horizontal distance gets smaller and
smaller. And the closer those two points
are together the more accurate our
estimate is for the slope at that point.
And we can make it perfect if we make h so
small - make h infinitely smaller by taking
the limit as h approaches zero. So we're
going to do a better and better
approximation until it's spot on. So the
secant line becomes the tangent line as
this point shifts toward x. By the time
it's the same point as x, this line is
accurate. It's the tangent line slope. So
the secant line becomes the tangent line.
The slope of the secant line becomes the
slope of the tangent line. In other words,
the limit of the secant slope is the
tangent line slope when it's only touching
at one point. And so this limit is the
slope of the tangent line. And surprise,
surprise that limit also defines the
derivative. This is the definition of the
derivative. So we have finally arrived at
the definition of the derivative.
So this is the definition of the derivative. When
that limit exists, this defines that
that's the derivative. And this notation
you just read as f'(x). It just means the
derivative of f. Also I should say because
that slope, tangent line slope, can be
always changing in different places -
it's variable. It can be variable
and that's why the derivative is a
function. It is itself a function just
like the original function also a function.
Just to reiterate the derivative tells
you the slope everywhere - how steep, rise
over run, slope, slope number, at the risk
of sounding like a broken record
derivative is basically slope. And to be
honest this might not really click and
make sense until you see it used in
something like physics where if you're
looking at something moving in time, the
position changing in time, a curved graph
would mean that the slope is always
changing. The velocity is changing and not
constant because there's an acceleration.
Whereas if you had a straight-line graph,
the slope would be the same everywhere
because the velocity is not changing
because there's no acceleration.
But anyway, this is the definition of the derivative.
Now the question is.. how do you use it?
OK, so say you have to
find the derivative using the definition
of the derivative or by the limit process,
this is what you use. You use the
definition of the derivative but for your
f(x) - whatever you're given. So to find
the derivative f'(x) it's going to be
equal to the limit as h approaches zero.
That's there out front. Limit as h
approaches zero. And then we fill in this
part of the formula. f(x) + h means take
your f, and in place of x, you use x + h,
like all of x + h in place of x everywhere
x appears. So let's do that.
The f(x + h) part... which looks like this
3 times (x + h)^2, instead of x^2, plus 12.
Then you subtract f(x), but you're
subtracting all of f(x) just as is
however the f(x) looks, but make sure
you use parenthesis when you do the
subtraction so you get the right signs. So
we have minus all of f(x). And then it's
all over h in the formula. Now we can just
do some algebra and simplify.
OK, so here's all the work to find the
derivative. Our answer was 6x but in the
work, I foiled. I distributed. I factored.
I canceled terms. At one point I factored
out an h so that it would cancel with the
bottom h. And in the end I took the limit.
And I was able to take the limit by
plugging in zero for h. And I got 6x which
is a beautiful, simple result after all
that work in algebra. The derivative is just 6x.
f'(x) is 6x. What that means is
that the derivative of this function
3x^2 + 12 is just 6x. And anywhere in
that function f, the slope would be the
number you get by using 6x at any instant,
any x. The slope is what you get from 6x
at that instant. So that's how you find
the derivative using the definition with
the limit. Just remember that... that's the
only tricky part is that this part
f(x) + h means
that instead of x, you use x + h. So
all of x + h in place of x. And at this
point, I should say, just FYI, as a public
service announcement, this way with the
definition and the limit is good and all.
It's correct and illustrative, but in
reality, it's pretty tedious. It's a lot
of extra algebra and in practice, that's
not really how most people take
derivatives. So if you're taking a lot of
derivatives, all this extra algebra is,
what's the word... a disaster. And there's a
faster, simpler way. And if nothing
explicitly says you have to find it using
the definition or by the limit process,
you can use the derivative rules, which
are a faster, simpler way. Much less extra
algebra. And I have a video on how to find
the derivative with the derivative rules.
So you can jump to that for that
explanation. So I hope that helped you
understand the derivative. I know calculus
is exactly what you want to be doing right
now. It's OK. You don't have to like math...
but you can like my video! So if you
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