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Differentiate.

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Differentiating polynomials is not so bad.
I mean for example, if I want to do

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you know differentiate some polynomial,
maybe 2x minus 4x cubed.

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plus 3x to the 10th, say.

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Well, all I've gotta do is remember my
rules for differentiating.

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You know I differentiate these sums and
differences

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by differentiating each term, and I can
differentiate a

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power just by bringing down the power and
subtracting 1, right.

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So the derivative of 2x is just 2.

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The derivative of 4x cubed is 12x squared.

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The derivative of 3x to the 10th is 30x to
the 9th.

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Turns out it's not too much worse to
differentiate a power series.

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Well, here's how you do it.
It's a theorem.

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Suppose I've got some power series.

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And I'm calling that f of x.
And big R is the

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radius of convergence of this power
series.

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So for any X between minus R and R, this
function f of

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x is defined to be the value of this
thing, convergence power series.

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Now here is the, the theorem then, the
derivative of

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this function f is this the sum n goes
from

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1 to infinity of n times a sub n times x
to the n minus 1 and if that looks

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mysterious, where is that coming from?

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Well that's just the derivative of a sub n
times x to the n.

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So this is telling you that if you want to

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differentiate a function, which was given
to you as a

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power series, well then the derivative is
just the sum

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of the derivatives of the terms of the
power series.

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You can differentiate term by term.
This new power series has the same

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radius of convergence as the old power
series and

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this power series for any value of X
between minus

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R and R is equal to the derivative of this

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function which is given to you as a power
series.

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At this point you might be wondering why
I'm even calling this a theorem.

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I mean what's the big deal?

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You might be thinking, or remembering,
that the derivative

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of the sum is the sum of the derivatives.

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So, what's the big deal?

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I mean,

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isn't there something that we already
know.

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The situation here, that the derivative of
a

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power series is the power series of the
derivative.

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It's actually way more subtle.

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Well the issue is that's not really what
we're talking about.

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It is true that the derivative of a sum is
just the sum of the derivatives, but what

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I am asking is whether the derivative of a
series is the series of the derivatives.

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That's really something more complicated.
Right?

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What's the definition of the series.

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Well it's the derivative of the limit of
the partial sums, and

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I'm wondering, is that the limit of the
sum of the derivatives?

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And although we do have a theorem that the
derivative of a sum is the

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sum of the derivatives, we don't have a
theorem, and it's not true in general.

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That the derivative of a limit is the
limit of

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the derivatives.
So it's a big deal.

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The fact that you can differentiate a
power series term by

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term, it's a theorem, I mean, that's
really something that's not obvious.

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But I can approve this theorem, but we are
going to make

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use of it and I think you'll find that
it's extraordinarily helpful.

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