WEBVTT
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Hi.
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Welcome back to recitation.
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In lecture you talked
about computing derivatives
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by definition.
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And one rule for
computing derivatives
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that Professor Jerison
mentioned but didn't prove
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was what's called the
constant multiple rule.
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So today I want to give
you a proof of that rule
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and show you a little bit of
geometric intuition for why
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it works.
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So the constant
multiple rule says
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that if you have a constant c
in a differentiable function, f
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of x, that the derivative of
the function c times f of x
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is equal to c times the
derivative of f of x.
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Just to do a quick example,
suppose that c were 3
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and f of x were the
function x squared,
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this says that the derivative
d by dx of 3 x squared
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is equal to 3 times
the derivative of d
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by dx of x squared.
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Now, this is good
because we already
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have a rule for computing
derivatives of powers of x.
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So this says we don't
need a special rule
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for computing multiples
of powers of x squared.
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We don't need to go back
to the limit definition
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to compute the derivative
of 3 x squared.
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We can just use the
fact that we know
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the derivative of x
squared in order to compute
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the derivative of 3 x squared.
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So in this case that
would work out to 6x.
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In this case.
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So it simplifies the number
of different computations
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you have to do.
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It reduces the number of
times we need to go back
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to the limit definition.
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So that's the use of the rule.
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Let's quickly talk
about its proof.
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The idea behind the proof is
you have these two derivatives
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and you want to show
that they're equal.
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Well, any time you have a
derivative, what it really
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means is it's the value of
some limit of some difference
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quotient.
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So in this case we have the
derivative d by dx of c times f
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of x by definition is
the limit of a difference
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quotient as delta x goes to 0
of-- so we take the function c
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times f of x and we
plug in x plus delta x
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and we plug in x and
we take the difference
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and we divide by delta x.
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So that's c times f of x
plus delta x minus c times
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f of x divided by delta x.
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Now you'll notice that here
both terms in the numerator
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have this constant
factor, c, in them.
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So we can factor that out.
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And I'll just pull it out in
front of this whole fraction
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so that this is
the limit as delta
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x goes to 0 of c
times the ratio f of x
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plus delta x minus f of x,
all quantity over delta x.
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Now, c is just some constant.
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This part depends on delta x.
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And on x, but on delta x.
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So as delta x goes
to zero, this changes
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while this stays the same.
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What that means is, so
as delta x goes to 0,
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this gets closer and
closer to something,
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the value of its limit.
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And c, you're just
multiplying it in,
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so c times-- the limit of c
times this is equal to c times
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whatever the limit of this is.
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If this is getting closer
and closer to some value,
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c times it is getting closer and
closer to c times that value.
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So this is equal to
c-- in other words,
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we can pull constant
multiples outside of limits.
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So this limit as
delta x-- c times
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the limit is delta x goes
to 0 of f of x plus delta x
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minus f of x over delta x.
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And this limit here
is just the definition
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of the derivative of f of x.
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So this is equal
to, by definition,
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c times d by dx of f of x.
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So we started with
the derivative
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of c times f of x and we
showed this is equal to c
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times the derivative of f of x.
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That's exactly what we wanted.
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So that proves the
constant multiple rule.
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We've now proved the
constant multiple rule--
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let me talk a little bit
about some geometric intuition
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for why this works.
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So I've got here, well so, you
know, let's take c equals 2,
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just for simplicity.
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So here I have a
graph y equals f of x,
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and I have also drawn the
graph, y equals 2f of x.
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The relationship
between these graphs
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is that y equals 2f of x is
what you get when you stretch
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the graph for y equals f of x
vertically by a factor of 2.
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So, you know, if it
passed through 0 before,
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it still passes through 0.
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But everywhere else,
if it was above 0,
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it's now twice as high.
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If it was below 0,
it's now twice as low.
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So if you think about what the
definition, what the derivative
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means in terms of this
graph geometrically,
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it's telling you
the limit-- sorry,
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the slope of a tangent line.
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Or in other words, the limit
of the slopes of secant lines.
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So if you look at
these two curves,
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say-- let's pick a
couple values of x, say,
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and then maybe x
plus delta x-- so
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if you look at the secant
lines for these two curves
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through those
points, what you see
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is that these two lines,
they have the same you know,
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so the slope of a line is
its rise over its run--
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so they have the
same run, that we
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are talking about the same
little interval, here.
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But this, in the function
that's scaled up,
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in the y equals
2f of x curve, we
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have that that the rise
everything has been stretched
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upwards by a factor of
two-- so the rise here is
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exactly double the rise here.
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So the slope of the secant line
is exactly double the slope
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of this secant line.
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And similarly,
the tangent line--
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just a limit of secant
lines-- has been stretched
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by that same factor of two.
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So the slope of the
tangent line is exactly
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twice the slope of the tangent
line for the other function.
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So the tangent line here
is exactly twice as steep
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as the tangent line here.
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Or in other words, the
derivative of this function
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is exactly twice the
derivative of that function.
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So that's just a
geometric statement
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of this very same
constant multiple rule
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that we stated algebraically
at the beginning
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and that we just proved.
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So that's that.