Differentiate.
Differentiating polynomials is not so bad. I mean for example, if I want to do
you know differentiate some polynomial, maybe 2x minus 4x cubed.
plus 3x to the 10th, say.
Well, all I've gotta do is remember my rules for differentiating.
You know I differentiate these sums and differences
by differentiating each term, and I can differentiate a
power just by bringing down the power and subtracting 1, right.
So the derivative of 2x is just 2.
The derivative of 4x cubed is 12x squared.
The derivative of 3x to the 10th is 30x to the 9th.
Turns out it's not too much worse to differentiate a power series.
Well, here's how you do it. It's a theorem.
Suppose I've got some power series.
And I'm calling that f of x. And big R is the
radius of convergence of this power series.
So for any X between minus R and R, this function f of
x is defined to be the value of this thing, convergence power series.
Now here is the, the theorem then, the derivative of
this function f is this the sum n goes from
1 to infinity of n times a sub n times x to the n minus 1 and if that looks
mysterious, where is that coming from?
Well that's just the derivative of a sub n times x to the n.
So this is telling you that if you want to
differentiate a function, which was given to you as a
power series, well then the derivative is just the sum
of the derivatives of the terms of the power series.
You can differentiate term by term. This new power series has the same
radius of convergence as the old power series and
this power series for any value of X between minus
R and R is equal to the derivative of this
function which is given to you as a power series.
At this point you might be wondering why I'm even calling this a theorem.
I mean what's the big deal?
You might be thinking, or remembering, that the derivative
of the sum is the sum of the derivatives.
So, what's the big deal?
I mean,
isn't there something that we already know.
The situation here, that the derivative of a
power series is the power series of the derivative.
It's actually way more subtle.
Well the issue is that's not really what we're talking about.
It is true that the derivative of a sum is just the sum of the derivatives, but what
I am asking is whether the derivative of a series is the series of the derivatives.
That's really something more complicated. Right?
What's the definition of the series.
Well it's the derivative of the limit of the partial sums, and
I'm wondering, is that the limit of the sum of the derivatives?
And although we do have a theorem that the derivative of a sum is the
sum of the derivatives, we don't have a theorem, and it's not true in general.
That the derivative of a limit is the limit of
the derivatives. So it's a big deal.
The fact that you can differentiate a power series term by
term, it's a theorem, I mean, that's really something that's not obvious.
But I can approve this theorem, but we are going to make
use of it and I think you'll find that it's extraordinarily helpful.
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