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Derivative Question

  1. Dec 15, 2009 #1
    I am studying this from a book I found online, and I need a little bit of help.

    1. The problem statement, all variables and given/known data
    Find the derivative when [itex]y(x)=\sin x[/itex]

    3. The attempt at a solution

    [itex]\frac{dy}{dx} \ = \ limit \ of \ \frac {\Delta y}{\Delta x} \ = \ \lim h \rightarrow 0 \ \frac{\sin(x+h)-\sin x}{h}[/itex]

    [itex]\sin(x+h)=\sin x \cos h + \cos x \sin h[/itex]

    [itex]\frac{\Delta y}{\Delta x} \ = \ \frac {\sin x \cos h + \cos x \sin h-\sin x}{h} \ = \ \sin x ( \frac{\cos h-1}{h}) + \cos x (\frac{\sin h}{h})[/itex]
  2. jcsd
  3. Dec 15, 2009 #2


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    Homework Helper

    now take the limit as h→0, what does sinh/h tend to? and what does (cosh-1)/h tend to?
  4. Dec 16, 2009 #3


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    Science Advisor

    Strictly speaking, how you do that depends upon what your definitions of "sine" and "cosine" are- and there are several possible. What definitions are you using?
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