Finding the Power Series of f'(x) from f(x) = x^2cos(2x)

[Magnawolf]

Homework Statement

Find the power series of f'(x), given f(x) = x²cos²(x)

Homework Equations

Correct me if I'm wrong

The Attempt at a Solution

Can I just take the derivative of the solution I got previously? If so, what's a good way to write the sequence out so I can easily make a series representation. Or is there a better approach to the problem?

 

[LCKurtz]

Looks good. I would multiply the $x^2/2$ into the parentheses and include it under the summation. Then differentiate it.

 

[Magnawolf]

My main problem is trying to figure out how to represent what I'm getting as a series. When I multiply in the x^2 and differentiate I get

x + x - 8x^3 + 4x^5 - (32/45)x^7...

I can't figure it out

 

[LCKurtz]

When you multiply the $\frac{x^2} 2$ in you have$$
\frac {x^2} 2 +\sum_{n=0}^\infty\frac{(-1)^n2^{2n}x^{2n+2}}{(2n)!}$$Just differentiate the first term as you did and differentiate under the sum and you will have your formula.

 

[Magnawolf]

When you say differentiate under the sum, do you mean differentiate each term in the sequence or do you mean you can actually take the derivative of the series equation? I tried googling and checking my textbook and I don't know how to take a derivative of a series in equation form, with the n's and such. If you can show me, that'd be great!

 

[HallsofIvy]

Yes, "differentiate under the sum" means "differentiate term by term". I don't know what you mean by "take the derivative of the equation"!

 

[LCKurtz]

When you say differentiate under the sum, do you mean differentiate each term in the sequence or do you mean you can actually take the derivative of the series equation? I tried googling and checking my textbook and I don't know how to take a derivative of a series in equation form, with the n's and such. If you can show me, that'd be great!

If you look at each term inside the sum, it just a constant times a power of $x$. Use the power rule.

 

[Magnawolf]

Oh okay, I got it now. Thanks for everything man!