[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]