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Homework Help: Derivative of f(x) = sinxcosx

  1. Mar 22, 2010 #1

    Ryuk1990

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    1. The problem statement, all variables and given/known data

    f(x) = sinxcosx

    2. Relevant equations

    Product Rule: f(x)g(x) = f(x)Dg(x) + g(x)Df(x)

    3. The attempt at a solution

    I got to sinx(-sinx) + cosxcosx

    The answer is supposed to be 0

    Would the next step be turning it into -(sinx)2 + (cosx)2?
     
    Ryuk1990, Mar 22, 2010
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  3. Mar 22, 2010 #2

    l'Hôpital

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    Your work is right.

    It can't be zero, because otherwise, you could integrate it and you'd obtain a constant, and clearly sinxcosx is not a constant. It's actually (sin 2x)/2
     
    l'Hôpital, Mar 22, 2010
  4. Mar 22, 2010 #3

    jbunniii

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    Your answer is right (both forms).

    Another way to solve this is to recognize that (as l'Hôpital said):

    [tex]\sin x \cos x = \frac{1}{2} \sin 2x[/tex]

    which has derivative

    [tex]\cos 2x[/tex]

    The answer looks different, but remembering one's trig identities pays dividends:

    [tex]\cos 2x = \cos^2 x - \sin^2 x[/tex]
     
    jbunniii, Mar 22, 2010
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