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Math Help - Implicit Differentiation Question

  1. #1
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    Implicit Differentiation Question

    Could someone please help me implicitly differentiate the following:

    3=(1/x)(y^3+2y^2)

    Thanks for any help you can provide!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by jj18 View Post
    Could someone please help me implicitly differentiate the following:

    3=(1/x)(y^3+2y^2)

    Thanks for any help you can provide!
    I would first rewrite the equation as this:

    3x=y^3+2y^2

    Differentiating both sides with respect to x, we get 3=3y^2\frac{\,dy}{\,dx}+4y\frac{\,dy}{\,dx}

    All you need to do know is group the \frac{\,dy}{\,dx} terms together and then solve for \frac{\,dy}{\,dx}.

    --Chris
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  3. #3
    MHF Contributor chiph588@'s Avatar
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     3x = y^3+2y^2

     \frac{d}{dx}(3x) = \frac{d}{dx}(y^3+2y^2)

     3\frac{d}{dx}(x) = \frac{d}{dx}(y^3)+2\frac{d}{dx}(y^2)

     3 = 3y^2\left(\frac{dy}{dx}\right)+4y\left(\frac{dy}{d  x}\right)
    (use the fact that  \frac{d}{dx}(f(y)) = f'(y)\frac{dy}{dx} by the chain rule)

     \frac{dy}{dx} = \frac{3}{3y^2+4y}

    now substitue  \frac{y^3+2y^2}{x} for 3, which yields

     \frac{dy}{dx} = \frac{y^3+2y^2}{3xy^2+4xy}
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  4. #4
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    Quick follow up question

    Thanks!

    One follow up question just to verify: The solution is not complete until making the substitution (1/x)(y^3+2y^2) for 3?
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Actually you don't have to, I dont really know why I did...

    Both are right answers
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