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U2S4V05 Introducing the Chain Rule.txt
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U2S4V05 Introducing the Chain Rule.txt
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#
# File: content-mit-18-01-1x-captions/U2S4V05 Introducing the Chain Rule.txt
#
# Captions for MITx 18.01.1x module [FL1swgyrsus]
#
# This file has 97 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
Let's walk through what you just thought about.
And we're going to add in some pictures
to hopefully give you a better sense of what it all means.
Okay.
So we know that h of x is f of g of x,
and we're interested in h prime of 2.
And we were told that g of 2 equals 9.
So I'm going to draw some number lines here.
g is taking 2 to 9.
And we know that the 2 was measured in seconds
and the 9 was in meters.
And then, we had f of 9 was five kilograms.
So we have an arrow here for f sending 9 to 5.
Now, h is the composition of these two things,
so h is starting at the top with two seconds and h of 2
is five kilograms.
So that was the picture before we started
changing some of these numbers.
But then, we got derivative involved.
So we know about g prime and f prime, so there they are.
We have g prime of 2 and f prime of 9,
which are the derivatives at the relevant points.
And we used them to think about what happens when our input up
at the top changes.
So we had x changing from 2 to 2.01.
So that's a delta x of 0.01, and we
want to know what happens when we apply g and then
f to this new x.
Well, this is exactly the sort of thing
that derivatives are meant to deal with.
So through linear approximation, we
know that if we take our delta x of 0.01 seconds
and we multiply it by the derivative of g at 2,
which is three meters per second,
we'll be getting 0.03 meters, which will be approximately
the change in g of x.
Now, of course, we can do the same thing for f.
So our 0.03 meters is the change in f's input.
And f prime at this point is four kilograms per meter,
so linear approximation for the change in the output
would be 0.03 meters times four kilograms per meter,
which is 0.12 kilograms.
And so that's what we estimate to be delta f.
But that's also delta h.
So as h's input changed by this 0.01 seconds,
its output changed by this 0.12 kilograms.
So we have a delta h and we have a delta x,
and that means that we can get a handle on the derivative of h.
h prime of 2 is going to be roughly this delta h divided
by this delta x, which is 0.12 kilograms
divided by 0.01 seconds, or 12 kilograms per second.
Let's take a step back and think about where
this 12 number came from.
So it came from this 3 up here and this 4 right here.
So we multiplied delta x by 3 to get delta g.
We multiplied that by 4 to get delta h.
So our h prime of 2 is really our f prime
of 9 times g prime of 2.
And notice that the units work out perfectly.
f prime is kilograms per meter.
g prime is meters per second.
Multiply those, and you're getting kilograms per second
for our derivative of h.
So this is really our complete answer.
The derivative of this composition
is equal to the product of the derivatives
of the functions that are the parts of the composition.
So h prime of 2 is f prime of 9 times g prime of 2.
Now, you could ask why it's f prime of 9, not f prime of 2.
But if you go back to our diagram,
you can see that the relevant input for f is this 9 meters.
That's what g is feeding into f.
So that's where we should be applying f prime, as well.
What we've just done here is a particular instance
of what we call the chain rule, and the reason
it's called the chain rule is the following.
We've got our function h that we've been discussing,
and it's the composition of these functions g and f.
So you can think of this as f and g forming two links
in this chain of functions.
And what we discovered is that when
you take the derivative of h, then you just take
the derivatives of the two links and you
multiply those together.
And that gives you the derivative
of the overall composition.
So what you can think about next is
what happens if h is not the composition of two functions,
but perhaps the composition of three functions,
so something like this where we have h equaling j of f of g.
And once you've done that, we'll come back
and talk about the chain rule in general.