[MUSIC] Remember, f prime of x is 
encoding how wiggling x affects f of x. 
We can see this by thinking about slopes 
of tangent lines as well. 
Here, I've drawn in orange the graph of 
some random function y=f(x). 
And this point, a f(a) I've drawn in red 
the tangent line to the graph. 
I want to know how is the output f 
affected when input moves from a to a+h. 
I want to know what happens when I wiggle 
the input over by adding h. 
Let me zoom in on this picture a little 
bit. 
Here, I've got this tangent line, and the 
slope of that tangent line is the 
derivative of the function at the point 
a. 
That means that this triangle can be 
understood, right? This slope being 
f'(a), this base being h means this 
height is h*f'(a). 
In order to get this slope as f'(a), rise 
over run better be equal to f'(a), this 
divided by this is f'(a). 
this is giving me some information about 
how wiggling the input will affect the 
output. 
If I move from a to a+h, well, on the 
tangent line m' moving up to this point 
which isn't so far off of the real value 
of the function up here. 
Now, if I made h really, really small, 
I'd be doing an even better job of 
staying close to the graph of the 
function when I follow the tangent line. 
Instead of starting with the function and 
trying to figure out the derivative, we 
can imagine that we know a little bit of 
information about the derivative and try 
to figure out something about the 
function. 
Let's make up a concrete example. 
Suppose that I've got some function f, 
and all I know is that its derivative is 
3x, and its value at 2, is 4. 
Just knowing this information without a 
rule for the function at this point, can 
I say anything about the function's value 
at, say 2.01? Yes, I can. 
Right? f(2.01), well that's f(2) plus how 
much the output changes when I go from 2 
to 2.01. 
Well, the output change is approximately 
something that I can compute from the 
derivative, 
right? 
The derivative is infinitesimally the 
ratio between output change and input 
change. 
So if I multiply by how much I change the 
input by the ratio of input change to 
output change, 
this should be approximately the true 
output change. 
Now, in this case, I know what these 
numbers are. 
0.01 times the derivative of f(2) is 6. 
Which means that this is about 4+0.06 
which is 4.06. 
And that's about what this function at 
2.01 is equal to. 
One more cat moves into the neighborhood. 
What happens to the mouse population? 
Right? A small change to one thing will 
affect something else. 
That's what the derivative is encoding. 
[MUSIC]