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Math Help - quick differentiation question about sin^2(x)

  1. #1
    Red
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    quick differentiation question about sin^2(x)

    Ignorant question here, but I was looking at this example i have, needing to take the derivative:

    f'(x) = sin^2(x) + cos(x)
    = 2sin(x)*d/dx(sin x) - sin(x)
    = 2sin(x)*cos(x) - sin(x)

    and so on... I'm wanting to know why the 2sin(x) isn't where the differentiation stops for sin^2(x)? there's the 2sin(x), and then the d/dx(sinx) added in as well. The cos(x) on the right side doesn't have any extra steps for it. Hope this makes sense, and i realize this is a silly question but I just wanted the clarification.
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  2. #2
    MHF Contributor Unknown008's Avatar
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    You can consider this function as follows:

    f(x) = (sin(x))^2 + cos(x)

    Then,

    f'(x) = 2(sin(x)).cos(x) - sin(x) = 2sin(x)cos(x) - sin(x)

    Just as the equation:

    y = (x^2+2)^2

    becomes:

    y' = 2(x^2 + 2). (2x) = 4x(x^2 + 2)
    Last edited by Unknown008; June 24th 2010 at 08:13 AM. Reason: Forgot a math tag
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    f(x)=sin^2(x)

    f'(x)=2sin(x)\cdot (sinx)'=2sin(x)\cdot cos(x)=sin(2x)
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  4. #4
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    Quote Originally Posted by Red View Post
    I'm wanting to know why the 2sin(x) isn't where the differentiation stops for sin^2(x)?
    Because it's not true that [LaTeX ERROR: Convert failed] . In general, [LaTeX ERROR: Convert failed] unless [LaTeX ERROR: Convert failed] The derivative of f at x is defined as the limit [LaTeX ERROR: Convert failed] , but unfortunately, many students (at school, anyway) understand it as [LaTeX ERROR: Convert failed] . The power rule is itself as a result of considering the function [LaTeX ERROR: Convert failed] at [LaTeX ERROR: Convert failed] , and using the definition of the derivative to calculate [LaTeX ERROR: Convert failed] as [LaTeX ERROR: Convert failed] . If I were to ever teach (school) calculus, it's one of those things I would write in giant letters somewhere above the board.
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    You can write only the words: "By definition"
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