Finding nth derivative of function

[TsAmE]

Homework Statement

For each n, let y^(n) denote the nth derivative of the function y = e^(x) cosx.

Can you suggest a general formula for y^(k), where k is an interger > 0? What is y^(2010)?

Homework Equations

None

The Attempt at a Solution

I got e^(x) (cosx - sinx) for y'

-2e^(x) sinx for y''

-2e^(x) (cosx + sinx) for y''' and

-4e^(x) cosx for y''''

which are right, but I don't see a pattern for the nth derivative, which could help me answer the question.

 

[Cyosis]

Start by looking at the even derivatives.

 

[lawtonfogle]

I would suggest doing another 4 derivatives, and then see if you can notice the pattern. Often times when I am looking for a pattern in data, more data really helps.

Is there a name for a function than takes in some natural number n and gives the n'th derivative?

 

[TsAmE]

Start by looking at the even derivatives.

I got:

y^(6) = 8e^(x) sinx

y^(8) = 16e^(x) cosx

but still don't see a pattern in comparison to:

-2e^(x) sinx for y''
-4e^(x) cosx for y''''

as it isn't alternating negative/positive sign or having a constant sign, but otherwise it alternating between cosx and sinx and is doubling

 

[Cyosis]

Take the derivative of cos x four times and you will see the pattern when it comes to the minus signs. Yes it is doubling you should be able to link this to 2^n.

 

[TsAmE]

But for y(2), y(4), y(6) and y(8) the signs are +, +, -, -. I can't see a pattern to represent these in an nth formula

 

[vela]

The cycle repeats every 4th derivative, so you might find it helpful to think about how to express y^(4m), y^(4m+1), y^(4m+2) and y^(4m+3). For example, you found that

y^(0) = e^x cos x = (-4)⁰ cos x
y^(4) = -4 e^x cos x = (-4)¹ cos x
y^(8) = 16 e^x cos x = (-4)² cos x

so you can say that y^(4m) = (-4)^m cos x.

 

[Cyosis]

But for y(2), y(4), y(6) and y(8) the signs are +, +, -, -. I can't see a pattern to represent these in an nth formula

No y(0),y(2),y(4),y(6) are +,-,-,+. Have you taken the derivative of the cosine four times yet?