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Using Derivative to give a general formula

  1. Oct 28, 2011 #1
    1. The problem statement, all variables and given/known data
    a) Find dn/dxn (xn) for n = 1,2,3,4,5. Give the general formula

    b) Give the general formula for dn+1/dxn+1 (xn)


    2. Relevant equations



    3. The attempt at a solution
    a) n = 1 : d/dx (x) = 1
    n = 2 :d2/dx2 (x2) = 2
    n = 3 :d3/dx3 (x3) = 6
    n = 4 :d4/dx4 (x4) = 24
    n = 5 :d5/dx5 (x5) = 120

    Notice that there is a recurring pattern:
    1,2,6,24,120→ multiply the derivative of the previous term by the existing term (2 times 1 = 2, 3 times 2 = 6, 4 times 6 = 24, 5 times 24 = 120)

    This is where i'm a little confused, i know what the pattern is and it is a bit hard to explain, however i wrote this out as the general formula:
    dn/dxn (xn) = n[dn-1/dxn-1 (xn-1)
    Would this be considered right? Or is the general formula suppose to be an arithmetic sequence or a geometric sequence?

    b) dn+1/dxn+1 (xn+1)
    n = 1 :d2/dx2 (x1) = 0
    n = 2 :d3/dx3 (x2) = 0
    n = 3 :d4/dx4 (x3) = 0
    n = 4 :d5/dx5 (x4) = 0
    n = 5 :d6/dx6 (x5) = 0
    So then the general formula is:
    dn+1/dxn+1 (xn+1) = 0
    Once again is this correct or would there be another sequence equation for this like the last one. I am fairly sure that there is not much of an issue of a sequence formula for part b because it will always result in zero but for part a im confused as to if my equation is correct and if it is not, what would the general formula be? Thank you.


    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 28, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I think everything you've said is right. In terms of writing a general formula, you might want to check out the definition of the 'factorial' function written "n!".
     
  4. Oct 28, 2011 #3

    Mark44

    Staff: Mentor

    The above looks fine. Another way to write the result is to use the factorial.

    d2/dx2 (x2) = 2 = 2!
    d3/dx3 (x3) = 6 = 3!
    d4/dx4 (x4) = 24 = 4!
    d5/dx5 (x5) = 120 = 5!

    Your sequence 1, 2, 6, 24, 120, ... is neither an arithmetic sequence, in which each term is some constant plus the previous term, or a geometric sequence, in which each term is some constant times the previous term.
     
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