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Homework Help: Finding nth derivative of function

  1. Apr 23, 2010 #1
    1. The problem statement, all variables and given/known data

    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)?

    2. Relevant equations


    3. 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.
  2. jcsd
  3. Apr 23, 2010 #2


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    Start by looking at the even derivatives.
  4. Apr 23, 2010 #3
    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?
  5. Apr 23, 2010 #4
    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
  6. Apr 23, 2010 #5


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    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.
  7. Apr 24, 2010 #6
    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
  8. Apr 24, 2010 #7


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    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.
  9. Apr 24, 2010 #8


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