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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.