Question

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Answer 1

this problem becomes a lot easier if you recognize that
g'(x) = -2x*sin(x^2) is the derivative of cos(x^2). Please verify that this is the case.

Answer 2

for g'(x) or g(x)?

Answer 3

It's the derivative of cos(x^2)-1

Answer 4

But I need to find a power series representation for g"(x) not g(x)

Answer 5

g'(x) i mean

Answer 6

do you know what series represent sinx?

Answer 7

I agree that you, jenn, are looking for the power series repr. of g'(x).
Please look at my previous post and respond to it if you can.

Answer 8

Yeah, I posted, that it's the derivative of g(x)=cos(x^2)-1.

Answer 9

Wondering why you're subtracting 1 from cos(x^2).
Look: if you want a power series for g(x)=cos(x^2), please look up the series for y=cos x. Next, everywhere you see x in that series, replace it with (x^2).

then differentiate the series. These resulting derivative of the series will be the series representation of g'(x)=-2x*sin(x^2).

Answer 10

I'm subtracting one from it because that's the way it was written in the problem.

Answer 11

I'm confused as to what series I should look up?

Answer 12

are you trying to find the power series representation for g(x), g'(x), or g''(x)?

Answer 13

g'(x)

Answer 14

which is -2x*sin(x^2)

Answer 15

well there are two way, since d/dx (cos(x^2) ) = -2x sin(x^2), find the power series for cos(x^2), and then differenticate it

Or, you can find power series for sin(x^2), then multiply the series by -2x

Answer 16

If I did the first way, how would I find the power series for cos(x^2)? I don't know how to do that

Answer 17

well, do you know power series for cos(x)? just replace x with x^2

Answer 18

oh okay

Answer 19

sourwing and I are both assuming that you have seen the derivation of power series for various functions before, and that you have a table of series for some common functions, such as cos x. Are we right, or ... ?

Answer 20

well, I just looked up the power series for cos(x) so..

Answer 21

So, assuming you have it, throw out every x and replace each with (x^2).

Answer 22

I did that and differentiated it too

Answer 23

Not sure if I got the right answer but thanks for helping :)

Answer 24

You're welcome. If you need further clarification, either ask questions here in the present discussion, or repost your question in the usual way.