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Homework Help: Find the derivative and the domain of the derivative (trig funtions)

  1. Nov 10, 2011 #1
    1. The problem statement, all variables and given/known data


    2. Relevant equations

    I am supposed to find the derivative, and the domain of it. The problem is that I cannot deal with the simplification of it.

    3. The attempt at a solution

    As far as I know, I am suppsosed to use f(x+h)-f(x)/h
    = sin(2(x+h)+5)-(sin(2x+5))/h

    I tried expanding what's inside, as in sin(2x+2h+5)-sin(2x+5)/h
    But that doesn't seem to lead to a solution
    Last edited by a moderator: Nov 10, 2011
  2. jcsd
  3. Nov 10, 2011 #2


    Staff: Mentor

    Are you certain that you have to use the limit definition of the derivative here? Have you learned the chain rule yet?
  4. Nov 10, 2011 #3
    I have learned the chain rule, and I am certain I am supposed to use the limit definition in this question
  5. Nov 10, 2011 #4


    Staff: Mentor

    OK, then I think you'll need to use the sum identity for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).

    You will probably need to use a couple of limit formulas as well:
    [tex]\lim_{h \to 0}\frac{sin(h)}{h} = 1[/tex]
    [tex]\lim_{h \to 0}\frac{cos(h) - 1}{h} = 0[/tex]
  6. Nov 10, 2011 #5


    User Avatar
    Homework Helper

    recall the trigonometric identity
    sin(2(x+h)+5)-(sin(2x+5))=2 sin(h) cos(h+2 x+5)
    or in general
    sin(A)-sin(B)=2 sin(A/2-B/2) cos(A/2+B/2)
    and use the facts that
    cosine is continuous
  7. Nov 11, 2011 #6

    Am I supposed to solve for when a --> 0?

    That's what I did.

    sin(2x+5)-sin(2a+5) = 2sin ((x-a)/2) cos ((x+a)/2)

    = sin (2x+5)-sin (5) = 2sin (x/2) cos (x/2)

    Then the x was = 0

    I am not sure I am on the right track. The derivative is cos(2x+5)(2) .-. I am not supposed to achieve a value. So was I supposed to keep the a?
  8. Nov 11, 2011 #7


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Actually sin(2x+5)-sin(2a+5) = 0

    You need to look at sin((2x+5)+h)-sin(2a+5) = 2sin (h/2) cos ((2x+5)+h/2) .
  9. Nov 11, 2011 #8

    Oh. Alright. Thank you very much :]
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