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Math Help - Derivative of Absolute Value

  1. #1
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    Derivative of Absolute Value

    hi guys can you please help me with this:

    b) if f(x) = |sin(x)|, find f'(x) and sketch the graphs of f and f'. Where is f not differentiable? you don't have to worry about the graphing part.
    a website said that the derivative is:
    \frac{cos(x)sin(x)}{|sin(x)|}

    can someone tell me how to get to that, if that's correct? I understand the rule is x/|x| right? Along with chain rule?

    there's also this one:
    c) If g(x) = sin|x|, find g'(x) and sketch the graphs of g and g'. Where is f not differentiable.
    the same website said g' is \frac{xcos|x|}{|x|} .
    How do I get there?
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  2. #2
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    f(x)=|\sin x|=\sqrt{\sin^2 x}
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  3. #3
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    Another way to do that is to note that, as long as sin(x) is positive that is just sin(x) which has derivative cos(x) while, if x< 0, it is -sin(x) which has derivative -cos(x).

    Then note that for any x except 0, \frac{x}{|x|} is +1 when x is positive, -1 when x is negative.
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  4. #4
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    I get it!
    (1/2(sin^2(x))^(-1/2))(2sin(x)cos(x))= cos(x)sin(x)/|sin(x)|

    but can you guys help me graph that derivative please. Like can you explain the logic of it:
    http://www.numberempire.com/graphingcalculator.php?functions=cos(x)*sin(x)%2Fa bs(sin(x))&xmin=0&xmax=6&ymin=-1&ymax=1

    I understand that it acts like the graph of cos(x) and that it's 0/0 at pi. But I don't get why it immediately starts repeating.
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