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Nth derivative

  1. Jan 15, 2006 #1
    nth derivative!!!!!

    I have a quick question...
    what is the nth derivative of cos(X)?? I have written out the derivatives up to the 4th one...it being cos(X) as well, but i don't see the pattern for the nth because it continues to change between sin and cos. I'm doing this for Taylor Series. Thanks for any help.
  2. jcsd
  3. Jan 15, 2006 #2
    Well you have f(x) = cos(x) right


    f'(x) = -sin(x) 1st derivative
    f"(x) = -cos(x) 2nd derivative
    f"'(x) = sin(x) 3rd derivative
    f""(x) = cos(x) 4th derivative.
    and it would repeat after this right...

    see the pattern for a given n the nth derivative of cosine x can only be one of those 4 choices right.

    so if n/4 has a remainder of 1 the nth derivative is -sin(x)
    if n/4 has a remainder of 2 the nth derivative is -cos(x)
    if n/4 has a remainder of 3 the nth derivative is sin(x)
    if n/4 has a remainder of 0 ( n is divisible by 4) then the nth derivative is cos(x).

    Does this help at all?
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