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Math Help - differenciate

  1. #1
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    differenciate

    Find the derivative of the function y= x sin (1/x)


    Differentiate: y= (u^6 - 2u^3 + 5) /u^2

    Thanks!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by new yorker View Post
    Find the derivative of the function y= x sin (1/x)


    Differentiate: y= (u^6 - 2u^3 + 5) /u^2

    Thanks!
    The first one is a combination of product and chain rule:

    y'=\sin\left(\frac{1}{x}\right)+x\cos\left(\frac{1  }{x}\right)\cdot\left(-\frac{1}{x^2}\right)=\color{red}\boxed{\sin\left(\  frac{1}{x}\right)-\frac{1}{x}\cos\left(\frac{1}{x}\right)}

    Does this make sense?

    ------------------------------------------------------------------------

    The second one isn't bad:

    y=\frac{u^6-2u^3+5}{u^2}=\frac{u^6}{u^2}-2\frac{u^3}{u^2}+\frac{5}{u^2}=u^4-2u+\frac{5}{u^2}

    Can you differentiate this one?

    --Chris
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  3. #3
    MHF Contributor
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    For the first question, use the product rule:

    y = x \sin \left(\frac{1}{x}\right)

    \frac{dy}{dx} = x \left(-\frac{1}{x^2} \cdot \cos \left(\frac{1}{x}\right)\right) + \sin \left(\frac{1}{x}\right)

    For the second question, divide through by u^2:

    y = \frac{u^6 - 2u^3 + 5}{u^2}

    y = u^4 - 2u + 5u^{-2}

    \frac{dy}{dx} = 4u^3 - 2 - 10u^{-3}
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