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Math Help - find dy/dx

  1. #1
    Super Member bigwave's Avatar
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    Cool find dy/dx

    find \frac{dy}{dx}
     <br /> <br />
y=x^3(5x^2+1)^{-\frac{2}{3}}<br />

    the answer in book is:
     <br />
\frac{1}{3}x^2(5x^2+1)^{-\frac{5}{3}}(25x^2+9)<br />

    I tried using the product and power rule on this
    but did not see how they got the 25x^2+9
    also, this is under the section for implicit differentiation but I only see the variable x in it.
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  2. #2
    MHF Contributor
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    hi bigwave,

    f'(x)=\left(5x^2+1\right)^{-\frac{3}{2}}3x^2+x^3\left(-\frac{2}{3}\right)\left(5x^2+1\right)^{-\frac{5}{2}}10x

    =\left(5x^2+1\right)^{-\frac{5}{2}}\left(5x^2+1\right)3x^2-\frac{20x^4}{3}\left(5x^2+1\right)^{-\frac{5}{2}}

    =\left(5x^2+1\right)^{-\frac{5}{2}}\left(15x^4+3x^2-\frac{20}{3}x^4\right)

    =\left(5x^2+1\right)^{-\frac{5}{2}}\left(\frac{25}{3}x^4+3x^2\right)

    =\left(5x^2+1\right)^{-\frac{5}{2}}\left(\frac{x^2}{3}\right)\left(25x^2+  9\right)
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