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Math Help - Derivative of Trigonometric Function

  1. #1
    Junior Member StarlitxSunshine's Avatar
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    Derivative of Trigonometric Function

    <br />
f(x) = \frac{cot x}{(1 + csc x)}<br /> <br />

    Find  f'(x)
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  2. #2
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    Quote Originally Posted by StarlitxSunshine View Post
    <br />
f(x) = \frac{cot x}{(1 + csc x)}<br /> <br />

    Find  f'(x)
    You know how to find the derivatives of cot(x) and csc(x) don't you? It's pretty easy if you know the derivatives of sine and cosine, so I won't go through it, I'll just write them down:

    \frac{d}{dx}cot(x)=-csc^2(x)

    \frac{d}{dx}csc(x)=-csc(x)cot(x)

    So to find f'(x), just apply the quotient rule.

    f'(x)=\frac{[1+csc(x)][-csc^2(x)]-cot(x)[-csc(x)cot(x)]}{[1+csc(x)]^2}

    =\frac{[1-csc^2(x)]+cot^2(x)csc(x)}{[1+csc(x)]^2}

    EDIT: The last line is WRONG. I didn't use the distributive property on the first term. When I mulitplied 1+csc(x) by -csc^2(x) I should have obtained -csc^2(x)-csc^3(x).
    Last edited by adkinsjr; November 5th 2009 at 06:12 PM.
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  3. #3
    Junior Member StarlitxSunshine's Avatar
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    Yup Yup :3 I know those derivatives.

    The thing is, I got the first step (using the quotient rule), but after that, I'm not sure how you simplified to get the answer ?

    Sorry for the trouble of explaining it twice >_<
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  4. #4
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    Quote Originally Posted by StarlitxSunshine View Post
    Yup Yup :3 I know those derivatives.

    The thing is, I got the first step (using the quotient rule), but after that, I'm not sure how you simplified to get the answer ?

    Sorry for the trouble of explaining it twice >_<
    In the numerator

    lol, the reason why is I simplified it wrong. Sorry about that. I'll fix it in a minute.



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  5. #5
    Junior Member StarlitxSunshine's Avatar
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    Ohh... I think I understand what you did :33

    But I have the answer to the question && it doesn't match up.

    The problem is that I can't get from the quotient rule step to the answer.

    Sorry for complicating it so much >_<

    The Answer:

    \frac{-csc x}{1+cscx}
    Last edited by StarlitxSunshine; November 5th 2009 at 06:17 PM. Reason: Put space to make it less cluttered :33
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