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Hi.

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Welcome back to recitation.

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In lecture you talked about
computing derivatives by

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definition.

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And one rule for computing
derivatives that Professor

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Jerison mentioned but didn't
prove was what's called the

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constant multiple rule.

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So today I want to give you a
proof of that rule and show

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you a little bit of geometric
intuition for why it works.

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So the constant multiple rule
says that if you have a

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constant c in a differentiable
function, f of x, that the

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derivative of the function c
times f of x is equal to c

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times the derivative
of f of x.

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Just to do a quick example,
suppose that c were 3 and f of

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x were the function x squared,
this says that the derivative

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d by dx of 3x squared 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 have a rule for

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computing derivatives
of powers of x.

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So this says we don't need a
special rule for computing

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multiples of powers of x, where
we don't need to go back

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to the limit definition
to compute the

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derivative of 3x squared.

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We can just use the fact that
we know the derivative of x

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squared in order to compute the
derivative of 3x 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 you

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have to do.

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It reduces the number of times
we need to go back to the

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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

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some difference quotient.

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So in this case we have the
derivative d by dx of c times

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f of x by definition is the
limit of a difference quotient

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as delta x goes to 0 of-- so we
take the function c times f

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of x and we plug in x plus delta
x and we plug in x and

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we take the difference and
we divide by delta x.

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So that's c times f of x plus
delta x minus c times f of x

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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 x goes to 0 of c times

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the ratio f of x plus delta x
minus f of x, all quantity

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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 while

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this stays the same.

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What that means is so as dealt
x goes to 0, this gets closer

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and closer to something,
the value of its limit.

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And c--

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you're just multiplying it in--
so c times, the limit of

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c times this is equal to
c times whatever the

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limit of this is.

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If this is getting closer and
closer to some value, c times

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it is getting closer and closer
to c times that value.

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So this is equal to c--

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in other words, we
can pull constant

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multiples outside of limits.

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So this limit as delta x, c
times the limit is delta x

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goes to 0 of f of x plus delta
x minus f of x over delta x.

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And this limit here is just
the definition of the

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derivative of f of x.

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So this is equal to, by
definition, c times d

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by dx of f of x.

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So we started with the
derivative of c times f of x

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and we showed this is
equal to c times the

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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 for

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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, and I have also

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drawn the graph, y
equals 2f of x.

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The relationship between these
graphs is that y equals 2f of

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x is what you get when you
stretch the graph for y equals

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f of x vertically by
a factor of two.

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So, you know, if it passed
through 0 before, it still

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passes through 0.

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But everywhere else, if
it was above 0, it's

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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

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derivative means in terms of
this graph geometrically, it's

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telling you the limit--
sorry-- the slope

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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, say--

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let's pick a couple values
of x, say, and then maybe

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x plus delta x--

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so if you look at the secant
lines for these two curves

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through those points, what you
see is that these two lines,

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they have the same, you know--
so the slope of a line is its

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rise over its run-- so they
have the same run, that we

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were talking about the same
little interval, here.

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But this, in the function that's
scaled up in the y

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equals 2f of x curve, we have
that that the rise--

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everything has been stretched
upwards by a factor of two--

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so the rise here is exactly
double the rise here.

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So the slope of the secant line
is exactly double the

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slope of this secant line.

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And similarly, the tangent
line-- just a limit of secant

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lines-- has been stretched by
that same factor of two.

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So the slope of the tangent
line is exactly twice the

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slope of the tangent line
for the other function.

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So the tangent line here is
exactly twice as steep as the

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tangent line here.

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Or in other words, the
derivative of this function is

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exactly twice the derivative
of that function.

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So that's just a geometric
statement of this very same

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constant multiple rule that we
stated algebraically at the

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beginning and that
we just proved.

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So that's that.

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