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Math Help - derivative of rational/trigonometric functions

  1. #1
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    derivative of rational/trigonometric functions

    I was hoping someone could check my solution.

    1. Calculate the derivative of: \frac{\cos x}{\ x-1}

    f(x)=(cosx)(1-x)^{-1}

    f'(x)=(cosx)'(1-x)^{-1} + (cosx)((1-x)^{-1})'

    f'(x)=-sinx(1-x)^{-1} + cosx(-1)(1-x)^{-2}(-1)

    f'(x)=\frac{\-sinx}{\ 1-x} + \frac {\ cosx}{x^2-2x+1}

    Thanks in advance.

    Sincerely,

    Raymond MacNeil
    Last edited by raymac62; August 25th 2011 at 06:24 PM. Reason: Spelling in title
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  2. #2
    Super Member TheChaz's Avatar
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    Re: derivative of rational/trigonometric functions

    The step in the middle had a typo or two, but it's ok.
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  3. #3
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    Re: derivative of rational/trigonometric functions

    Oh, you mean I forgot to make 1 a negative exponent right? I see.
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  4. #4
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    Re: derivative of rational/trigonometric functions

    Quote Originally Posted by raymac62 View Post
    I was hoping someone could check my solution.

    1. Calculate the derivative of: \frac{\cos x}{\ x-1}

    f(x)=(cosx)(1-x)^{-1}

    f'(x)=(cosx)'(1-x)^{-1} + (cosx)((1-x)^{-1})'

    f'(x)=-sinx(1-x)^{-1} + cosx(-1)(1-x)^{-2}(-1)

    f'(x)=\frac{\-sinx}{\ 1-x} + \frac {\ cosx}{x^2-2x+1}

    Thanks in advance.

    Sincerely,

    Raymond MacNeil
    The Quotient Rule is quite straightforward...

    \displaystyle \begin{align*} \frac{d}{dx}\left(\frac{\cos{x}}{x - 1}\right) &= \frac{(x - 1)\frac{d}{dx}(\cos{x}) - \cos{x}\frac{d}{dx}(x - 1)}{(x - 1)^2} \\ &= \frac{-(x - 1)\sin{x} - \cos{x}}{(x - 1)^2} \end{align*}
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