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Derivative of a sinusoidal function. Need some Expertise!

  1. Mar 26, 2012 #1
    Hey forum, thanks in advance to anyone who can explain this to me!
    1. The problem statement, all variables and given/known data
    Find the derivative of each of the following:
    h(x) = 3e^sin(x + 2)

    3. The attempt at a solution

    My friend and I worked through this together and solved for:
    h’(x) = (3e^sin(x + 2))(cos(x + 2))

    Using my graphing calculator I have already confirmed this answer..


    2. Relevant equations

    What I need help with is just a little bit of light being shed on the concept..

    My book says : the chain rule says the derivative of u^(-1) is (-1)u^(-2) u'
    so I get that..
    my book also says: the derivative of the function f(x)=a^g(x), where a is a positive constant, is given by f'(x)=a^g(x) lna g'(x)

    so applying to the question above.. isnt 3e a positive constant..

    so my question basically is.. when working this problem through..
    is there a middle step where a 'lne' is established?

    I realize lne = 1 anyway.. but I'm just wondering for the sake of making sure I thoroughly understand.

    and if there is a middle step involving the natural logarithm(ln) why wouldnt it be ln3e
    I realize its not because that would change the answer that I already confirmed but I'm just struggling to understand why not..
    cuz if a mid step with ln is involved.. 3 * e is a positive constant I thought so I'd figure it would be ln3e :S

    or another option is its lne * 3 which still equals 1..

    anyway, I'm rambling now so if someone could please explain this to me I would really appreciate it!:confused:
     
  2. jcsd
  3. Mar 26, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    The function is [itex] 3 \,e^{\sin(x+2)}, [/itex] with the '3' just being a constant 'outside' all the other computations. If you wanted to exponentiate 3e you would need to write [itex] (3e)^{\sin(x+2)}.[/itex]

    RGV
     
  4. Mar 26, 2012 #3
    h(x) = 3*(e^sin(x + 2))

    If f(x)= c*g(x) then f'(x)= c*g'(x) ( you can verify this by product rule).

    In this equation c= 3 and g(x) = e^sin(x+2).

    You are confusing H(x) = (3e)^sin(x+2) and h(x)= 3*e^sin(x+2). They are two different functions with two different derievatives.


    The answer you intially got is correct. Do you understand why?
     
  5. Mar 26, 2012 #4
    :D ohhhh yes I do.. damn that seemed really obvious >.>
    so we are using:

    the derivative of the function f(x)=a^g(x), where 'a' is a positive constant, is given by f'(x)=a^g(x) lna g'(x)

    and for the question h(x) = 3e^sin(x + 2)
    lne is established, however, simplified out later because it's value is 1
    Right?

    Thanks to both of you for your help :D
    who knows how many marks you just saved me on my upcoming test:eek:

    also.. if we slightly altered the question to:
    h(x) = 3e^sin(5x + 2)

    would the answer be:
    h’(x) = (15e^sin(5x + 2))(cos(5x + 2))
     
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