# Finding nth derivative of function

## 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)?

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 dont see a pattern for the nth derivative, which could help me answer the question.

## Answers and Replies

Cyosis
Homework Helper
Start by looking at the even derivatives.

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?

Start by looking at the even derivatives.

I got:

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

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

but still dont see a pattern in comparison to:

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

as it isnt alternating negative/positive sign or having a constant sign, but otherwise it alternating between cosx and sinx and is doubling

Cyosis
Homework Helper
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.

But for y(2), y(4), y(6) and y(8) the signs are +, +, -, -. I cant see a pattern to represent these in an nth formula

vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
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) = ex cos x = (-4)0 cos x
y^(4) = -4 ex cos x = (-4)1 cos x
y^(8) = 16 ex cos x = (-4)2 cos x

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

Cyosis
Homework Helper
But for y(2), y(4), y(6) and y(8) the signs are +, +, -, -. I cant 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?