Last time, in our second section, we talked
about limits and continuity, which gave us
a great foundation for today’s topic, derivatives.
We’ll be wrapping our heads around the first
of two fundamentally important questions in
calculus: how to find the rate of change of
a function at a point.
Before we go further, let’s talk about what
we mean by the rate of change of a function
at a point.
We all know how to find the rate of change
of a line; it’s just the slope of the line.
For example, picture the graph of 2x+2.
We can tell just by looking at the equation
that the slope of the graph is 2.
And it doesn’t matter where we are on the
graph of the function, the slope is 2 everywhere.
But what about this graph?
What is the slope, or rate of change of this
function at this point, or this one?
What if we need to know the slope at both
of these points?
How can we find the slope of a curved surface?
Well, we don’t, really.
We’ll actually find the equation of the
line tangent to the curve, that passes through
that point on the graph because remember,
we already know how to find the slope of a
line.
Let me take a half a step back, and talk for
a brief moment about something I just said:
the line tangent to the curve.
What does it mean for a line to be tangent
to a curve?
It means that the line just barely skims the
graph, intersecting it at exactly one point.
Think of the tangent line in contrast to a
secant line, which cuts through the graph
at one or more points.
Now imagine a secant line, intersecting the
graph at two points.
As I move this second point closer to the
tangent point, the distance between the points
of intersection becomes smaller and smaller
and the secant and tangent lines become more
and more similar.
Eventually, the distance between the points
will become infinitely small, until eventually
the two points come together at a single point,
and the line no longer intersects the function
at two points, but instead only at one, and
this is the tangent line to the curve at that
point.
So in order to calculate the slope of the
function at a point, we need to find the slope
of the tangent line, and therefore obviously
the equation of the tangent line.
How can we do this?
Well, recall from algebra that to find the
equation of a line, we either need two points,
or a point and a slope.
We know that the slope is the very thing we’re
trying to solve for, so we’re going to have
to calculate the equation of the line using
two points.
We already have one point; it’s the point
where the function and the tangent line intersect
each other.
So we just need one more point.
Which point can we pick?
Well, since we currently only have one equation
on the table, the equation of the function
itself, we can only pick points on the graph
of the function.
So, let’s pick another point on the graph;
one that’s relatively close to our tangent
point.
Using these two points, we can find the equation
of the secant line that connects the two,
and call that an estimate of the equation
of the tangent line.
But of course, an estimate isn’t good enough.
We want the exact equation of the tangent
line.
To get a better estimate, we can move this
second point closer and closer to the tangent
point.
The closer we move the points together, the
more identical the secant and tangent lines
become.
You might be starting to see by now that,
if we eventually decrease the distance between
these points all the way to zero, we’ll
have only one point, and that will be the
tangent point.
To model this mathematically, we’ll have
to borrow an equation we learned way back
in algebra, for the average rate of change
of a function between two points.
(a,f(a)) is our tangent point, and (b,f(b))
is our second point.
Using these two points, we can calculate the
average rate of change of the function between
them.
Remember though that we’re trying to find
the instantaneous rate of change at the tangent
point.
If we start moving (b,f(b)) closer and closer
to (a,f(a)), and we continue calculating the
average rate of change between the closer
and closer points, we’ll start to see the
average rate of change approach a specific
value.
The value it’s approaching is the instantaneous
rate of change, also known as the slope of
the tangent line, which is the slope of the
function at the tangent point.
This brings us to the definition of the derivative.
Let’s modify slightly the formula for average
rate of change.
Instead of using a and f(a) we’ll use x
and f(x), and we’ll call the distance between
the tangent point and the secant point h.
Now we can simplify the function by canceling
the x’s in the denominator.
Then, thinking again about the tangent line
and borrowing what we learned last time about
limits, we can take the limit as h goes to
zero, and this will give us the slope of the
tangent line.
In other words, as we bring those two points
closer and closer together, h, which is the
distance between the points, gets smaller
and smaller, until eventually it becomes 0,
and then we have the slope of the tangent
line, or the instantaneous rate of change
of the function at the tangent point.
Before we wrap up today, let’s take a look
at one example.
The problem says we need to use the definition
of the derivative to find the instantaneous
rate of change of the function f(x)=x/(1-2x)
when x=1.
You can see that the definition has more or
less two components: f(x+h) and f(x).
We already have f(x); it’s our original
function.
To calculate f(x+h), we’ll take all of the
x’s out of our original equation, and put
an (x+h) in their place.
Now we can take f(x) and f(x+h) and put them
both into the definition.
Now we need to start simplifying.
The simplification process will be different
depending on the original function.
In this case, we have to find a common denominator
in the numerator of our function.
Eventually, once we’ve fully simplified,
we’ll take the limit as h goes to zero.
In other words, we’ll plug in 0 for h, and
any term involving h will disappear from our
derivative function.
What we’re left with is the derivative of
the original function, which we denote as
f prime of x.
To find the slope of the function when x=1,
we just plug 1 into the derivative function,
and we see that the slope of the function
is equal to 1, or put another way, the instantaneous
rate of change of the function at x=1 is 1.
Next time we’re going to expand on what
we’ve learned about the concept and definition
of the derivative to talk about some better
techniques we can use to calculate derivatives.
I’ll see you then.
