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[MUSIC] Remember, f prime of x is 
encoding how wiggling x affects f of x. 

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We can see this by thinking about slopes 
of tangent lines as well. 

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Here, I've drawn in orange the graph of 
some random function y=f(x). 

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And this point, a f(a) I've drawn in red 
the tangent line to the graph. 

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I want to know how is the output f 
affected when input moves from a to a+h. 

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I want to know what happens when I wiggle 
the input over by adding h. 

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Let me zoom in on this picture a little 
bit. 

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Here, I've got this tangent line, and the 
slope of that tangent line is the 

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derivative of the function at the point 
a. 

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That means that this triangle can be 
understood, right? This slope being 

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f'(a), this base being h means this 
height is h*f'(a). 

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In order to get this slope as f'(a), rise 
over run better be equal to f'(a), this 

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divided by this is f'(a). 
this is giving me some information about 

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how wiggling the input will affect the 
output. 

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If I move from a to a+h, well, on the 
tangent line m' moving up to this point 

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which isn't so far off of the real value 
of the function up here. 

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Now, if I made h really, really small, 
I'd be doing an even better job of 

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staying close to the graph of the 
function when I follow the tangent line. 

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Instead of starting with the function and 
trying to figure out the derivative, we 

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can imagine that we know a little bit of 
information about the derivative and try 

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to figure out something about the 
function. 

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Let's make up a concrete example. 
Suppose that I've got some function f, 

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and all I know is that its derivative is 
3x, and its value at 2, is 4. 

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Just knowing this information without a 
rule for the function at this point, can 

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I say anything about the function's value 
at, say 2.01? Yes, I can. 

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Right? f(2.01), well that's f(2) plus how 
much the output changes when I go from 2 

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to 2.01. 
Well, the output change is approximately 

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something that I can compute from the 
derivative, 

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right? 
The derivative is infinitesimally the 

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ratio between output change and input 
change. 

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So if I multiply by how much I change the 
input by the ratio of input change to 

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output change, 
this should be approximately the true 

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output change. 
Now, in this case, I know what these 

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numbers are. 
0.01 times the derivative of f(2) is 6. 

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Which means that this is about 4+0.06 
which is 4.06. 

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And that's about what this function at 
2.01 is equal to. 

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One more cat moves into the neighborhood. 
What happens to the mouse population? 

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Right? A small change to one thing will 
affect something else. 

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That's what the derivative is encoding. 
[MUSIC]