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[MUSIC] We want to capture precise 
information about how wiggling the input 

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effects the output. 
Here's the difinition of derivative that 

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will allow us to do exactly that. 
The derivative of f at the point x is 

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defined to be this limit. 
The limit of f(x+h)-f(x)/h as h 

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approaches 0. 
Now, when this limit exists, I'm going to 

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say the function is differentiable, 
alright? 

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If the derivative of f exists at the 
point x, I'm going to to say the function 

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is differentiable at that point x. 
Sometimes, you'll see different 

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definitions of the derivative. 
Here's an equivalent one. 

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The derivative of f at x could 
equivalently be defined as this limit, 

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the limit as w approaches x of f(w)-f(x) 
all over w-x. 

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Now, how does this definition relate to 
our original definition of derivative? 

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The derivative of f at x is this limit 
involving h. 

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Well, look. 
In both cases, the numerators are 

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measuring how the output is changing. 
Here, I'm plugging in a nearby input and 

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I'm looking at how much the output 
changed compared to the output at x. 

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And h is measuring exactly how much that 
input changed by. 

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Here, I'm again measuring the difference 
of two output values. 

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Here, w is my new output value, which is 
close to x. 

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Alright, down here, h is just measuring 
how much I wiggled x by. 

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Here, w is actually just some nearby 
value of x. 

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And in both cases, the denominator is 
measuring how much the input was changed. 

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Here, h is exactly how much the input was 
changed by. 

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Here, w-x is measuring how much the input 
changed by. The other thing that makes 

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these definitions sometimes a little bit 
tricky is that people will give you a 

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definition that's not at the point of x. 
For instance, here's a definition of the 

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derivative of f at the point a. 
It's the limit of f(x)-f(a)/x-a as x 

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approaches a. 
You can compare the first and the third 

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definitions here. 
This first one has a w and an x and w is 

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approaching x to get the derivative at x. 
This bottom one, I'm trying to compute 

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the derivative at a and x is approaching 
a, alright? So, the roles of w and x and 

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the role of x and a are somehow analogous 
here. 

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Now, think back to when we were talking 
about continuity last week. 

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We started out with a definition of 
continuity at a single point and then we 

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expanded that definition to be continuity 
on a whole interval. 

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We played the same game with the 
derivative. Here we go. 

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If the derivative of f exists at x, 
whenever x is between a and b, but not at 

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a, or at b, we won't worry about that, 
just whenever x is between a and b. 

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And if this happens, then, we say that f 
is differentiable on the interval (a,b). 

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So, as a little bit of a warning here, 
this is not a point. 

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This is an interval. 
It's all the numbers between a and b, 

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not including a, 
not including b. 

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Now, contrast this with continuity, 
when we talked about continuity on an 

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interval, 
I also had separate definitions for 

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continuity and closed intervals or 
half-open intervals. 

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But that doesn't really make so much 
sense for the derivative. 

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here's why. 
The derivative is measuring how much 

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wiggling x affects f(x). 
And if I'm standing in the middle of an 

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interval, I can wiggle x. 
Even if I'm standing pretty close to b or 

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pretty close to a, I can always wiggle 
just a little bit to the left and a 

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little bit to the right, no matter how 
close I'm standing to a or how close I'm 

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standing to b, unless I'm standing, say, 
at the point b. 

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If I'm standing right at b, I can wiggle 
to the left. 

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But I can't wiggle at all to the right 
without walking right outside of the 

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interval. 
So, in light of this, I don't really want 

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to talk about differentiability on closed 
intervals. 

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I only want to talk about 
differentiablity when I can honestly talk 

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about wiggling the input and that's only 
true on open intervals. 

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There's plenty of other subtleties to 
this. 

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If I differentiate a function which is 
differentiable on a whole interval, then 

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I get a new function, f'. Specifically, 
the derivative of f at the point x will 

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be written like this, f with this little 
tick mark, and we're going to pronounce 

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that prime, f'(x). 
The point here is that I can define the 

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derivative, right, as this limit of this 
quotient. 

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And when you think of it this way, if 
this is really a function, right? This is 

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a rule that defines some new function, so 
I can regard f'(x) not just as a specific 

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values that I get by plugging specific 
values of x, but honestly as a function. 

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Alright. 
This thing is the sort of thing I can 

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plug any value of x into and see what I 
get out. 

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It's a function that somehow derived from 
f. 

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Maybe hence, the name derivative. 
There's a whole bunch of different 

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notation that you're going to see for the 
derivative in the wild. 

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Here are some of these. 
Alright, the derivitive of f at the point 

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x might be written as f'(x), like we've 
just been seeing. 

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Or it might be written d/dxf(x) or 
Dxf(x), or a bunch of other things, 

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right? 
Lots of different people have their 

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favorite notation for these. 
But really, this f' notation and this 

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d/dx notation are what we're going to be 
using in this course. 

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There are upsides and downsides to these 
various choices of notation. 

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For instance, here's a huge upside to 
this d/dx notation. 

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It's really emphasizing that the 
derivative is a ratio, right? It looks 

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like df, the change in the output, dx, 
the change in the input, somehow 

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revealing a little bit of how the 
derivative is actually defined. 

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The f'(x) notation doesn't emphasize that 
the derivative is a ratio, but it does 

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emphasize the derivative is a function. 
When we use the f'(x) notation, at least 

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you can tell this thing is a function. 
You've clearly labeled the input, x. 

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Differentiating gives a new function, 
even the name suggests that. 

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The derivative is somehow derived from 
the original function. 

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And if the derivative is now a function, 
you can then differentiate the 

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derivative. 
And differentiate that, and keep on 

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going. 
And dig deeper and deeper and deeper, 

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trying to uncover more and more secrets 
about the original function, 

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f. 
All of that is yet to come. 

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But the point is just that there's so 
much yet to explore. 

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[MUSIC]