[MUSIC] We want to capture precise information about how wiggling the input
effects the output. Here's the difinition of derivative that
will allow us to do exactly that. The derivative of f at the point x is
defined to be this limit. The limit of f(x+h)-f(x)/h as h
approaches 0. Now, when this limit exists, I'm going to
say the function is differentiable, alright?
If the derivative of f exists at the point x, I'm going to to say the function
is differentiable at that point x. Sometimes, you'll see different
definitions of the derivative. Here's an equivalent one.
The derivative of f at x could equivalently be defined as this limit,
the limit as w approaches x of f(w)-f(x) all over w-x.
Now, how does this definition relate to our original definition of derivative?
The derivative of f at x is this limit involving h.
Well, look. In both cases, the numerators are
measuring how the output is changing. Here, I'm plugging in a nearby input and
I'm looking at how much the output changed compared to the output at x.
And h is measuring exactly how much that input changed by.
Here, I'm again measuring the difference of two output values.
Here, w is my new output value, which is close to x.
Alright, down here, h is just measuring how much I wiggled x by.
Here, w is actually just some nearby value of x.
And in both cases, the denominator is measuring how much the input was changed.
Here, h is exactly how much the input was changed by.
Here, w-x is measuring how much the input changed by. The other thing that makes
these definitions sometimes a little bit tricky is that people will give you a
definition that's not at the point of x. For instance, here's a definition of the
derivative of f at the point a. It's the limit of f(x)-f(a)/x-a as x
approaches a. You can compare the first and the third
definitions here. This first one has a w and an x and w is
approaching x to get the derivative at x. This bottom one, I'm trying to compute
the derivative at a and x is approaching a, alright? So, the roles of w and x and
the role of x and a are somehow analogous here.
Now, think back to when we were talking about continuity last week.
We started out with a definition of continuity at a single point and then we
expanded that definition to be continuity on a whole interval.
We played the same game with the derivative. Here we go.
If the derivative of f exists at x, whenever x is between a and b, but not at
a, or at b, we won't worry about that, just whenever x is between a and b.
And if this happens, then, we say that f is differentiable on the interval (a,b).
So, as a little bit of a warning here, this is not a point.
This is an interval. It's all the numbers between a and b,
not including a, not including b.
Now, contrast this with continuity, when we talked about continuity on an
interval, I also had separate definitions for
continuity and closed intervals or half-open intervals.
But that doesn't really make so much sense for the derivative.
here's why. The derivative is measuring how much
wiggling x affects f(x). And if I'm standing in the middle of an
interval, I can wiggle x. Even if I'm standing pretty close to b or
pretty close to a, I can always wiggle just a little bit to the left and a
little bit to the right, no matter how close I'm standing to a or how close I'm
standing to b, unless I'm standing, say, at the point b.
If I'm standing right at b, I can wiggle to the left.
But I can't wiggle at all to the right without walking right outside of the
interval. So, in light of this, I don't really want
to talk about differentiability on closed intervals.
I only want to talk about differentiablity when I can honestly talk
about wiggling the input and that's only true on open intervals.
There's plenty of other subtleties to this.
If I differentiate a function which is differentiable on a whole interval, then
I get a new function, f'. Specifically, the derivative of f at the point x will
be written like this, f with this little tick mark, and we're going to pronounce
that prime, f'(x). The point here is that I can define the
derivative, right, as this limit of this quotient.
And when you think of it this way, if this is really a function, right? This is
a rule that defines some new function, so I can regard f'(x) not just as a specific
values that I get by plugging specific values of x, but honestly as a function.
Alright. This thing is the sort of thing I can
plug any value of x into and see what I get out.
It's a function that somehow derived from f.
Maybe hence, the name derivative. There's a whole bunch of different
notation that you're going to see for the derivative in the wild.
Here are some of these. Alright, the derivitive of f at the point
x might be written as f'(x), like we've just been seeing.
Or it might be written d/dxf(x) or Dxf(x), or a bunch of other things,
right? Lots of different people have their
favorite notation for these. But really, this f' notation and this
d/dx notation are what we're going to be using in this course.
There are upsides and downsides to these various choices of notation.
For instance, here's a huge upside to this d/dx notation.
It's really emphasizing that the derivative is a ratio, right? It looks
like df, the change in the output, dx, the change in the input, somehow
revealing a little bit of how the derivative is actually defined.
The f'(x) notation doesn't emphasize that the derivative is a ratio, but it does
emphasize the derivative is a function. When we use the f'(x) notation, at least
you can tell this thing is a function. You've clearly labeled the input, x.
Differentiating gives a new function, even the name suggests that.
The derivative is somehow derived from the original function.
And if the derivative is now a function, you can then differentiate the
derivative. And differentiate that, and keep on
going. And dig deeper and deeper and deeper,
trying to uncover more and more secrets about the original function,
f. All of that is yet to come.
But the point is just that there's so much yet to explore.
[MUSIC]