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U1S4V05 Delta notation for the derivative.txt
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U1S4V05 Delta notation for the derivative.txt
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#
# File: content-mit-18-01-1x-captions/U1S4V05 Delta notation for the derivative.txt
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# Captions for MITx 18.01.1x module [sthOiDRu6cY]
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# This file has 51 caption lines.
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# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
So here we have our original formula for the derivative.
But it's sometimes going to be useful to have
a different notation for it.
And that's what we're going to develop in this video.
So let's remember what all of these parts of this definition
meant.
So we have our function f and x and b
are representing different input values for our function.
So here they are.
And they correspond to these points on the graph.
Now the b minus x in the denominator
is referring to this horizontal distance.
And the f of b minus f of x in the numerator, that's
this vertical distance.
So the quotient is a vertical distance over a horizontal one,
and so it's the slope of this secant line.
And then we take the limit as this point moves towards x,
and that's our derivative.
OK.
So that's our original formula.
For our new notation, we're going
to focus not on this point b but on this gap.
So this b minus x, it's the difference in x values.
So we can denote it by delta x, the change
in x or the difference in x.
So remember that this delta x is a single number.
It's not delta times x or anything like that.
It's a single number.
It represents this difference between the two x values.
So this delta x is the new way that we're
going to write our denominator.
For the numerator, well, this numerator
is the difference in f values.
So one way to represent it would be with delta f.
We could also write it in terms of delta x.
So we know from down here that b is the same thing
as x plus delta x.
So instead of f of b minus f of x,
we can put f of x plus delta x minus f of x.
So our last question is what do we do with this limit?
So we're going to have the limit as something goes to something.
But what is it?
So what we want you to do is to take a moment
and to think about what that's going
to be using this new notation.
And when you're done, we'll come back
and we'll do an example of calculating the derivative
using this new notation.
All right?