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Math Help - Implicit differentiation

  1. #1
    Junior Member arcticreaver's Avatar
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    Implicit differentiation

    Hello,

    i have a problem

    x^2y^3+2y=6 solve using implicit differentiation.

    so am i suppose to take the derivative first?

    which would be 3x^2y^2+2xy^3-2=0?
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  2. #2
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    Re: Implicit differentiation

    Hey arcticreaver.

    What exactly are you trying to solve?
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  3. #3
    Junior Member arcticreaver's Avatar
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    Re: Implicit differentiation

    oh sorry, find dy/dx using implicit differentiation
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  4. #4
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    Re: Implicit differentiation

    Ok that makes more sense.

    Remember two rules: d/dx f(x) = f'(x) and d/dx f(u(x)) = u'(x)*f'(u(x)) (let u(x) = y). The rest includes the product, quotient and power rules.
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  5. #5
    Junior Member arcticreaver's Avatar
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    Re: Implicit differentiation

    can anyone work out the problem?
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  6. #6
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    Re: Implicit differentiation

    Quote Originally Posted by arcticreaver View Post
    can anyone work out the problem?
    Yes we can. The question is, can you? That's the whole point of the site, to give you guidance so that YOU can solve your problems.

    If the answer to that question is no, then what have you tried and where are you stuck?
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  7. #7
    MHF Contributor MarkFL's Avatar
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    Re: Implicit differentiation

    We are given the implicit relationship:

    x^2y^3+2y=6

    Now, using the product and chain rules, we find:

    x^2\left(3y^2\frac{dy}{dx} \right)+2x(y^3)+2\frac{dy}{dx}=0

    Now, solve for \frac{dy}{dx}
    Last edited by MarkFL; October 11th 2012 at 08:43 AM.
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  8. #8
    Junior Member arcticreaver's Avatar
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    Re: Implicit differentiation

    thanks for writing that out MarkFL2.

    dy/dx = -2/ x^2(6y^2)+2xy^3?
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  9. #9
    MHF Contributor MarkFL's Avatar
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    Re: Implicit differentiation

    That's incorrect...in general you want to arrange with the terms having dy/dx as a factor on one side and the other terms on the other. Then factor out dy/dx, and divide through by the other factor to isolate dy/dx. Give this another try...
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  10. #10
    Junior Member arcticreaver's Avatar
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    Re: Implicit differentiation

    okay so if i am understanding this correctly, it would be - x^2(6y^2)+2 / 2x(y^3)?
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  11. #11
    MHF Contributor MarkFL's Avatar
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    Re: Implicit differentiation

    Let's go back to:

    x^2\left(3y^2\frac{dy}{dx} \right)+2x(y^3)+2\frac{dy}{dx}=0

    Clean this up some:

    3x^2y^2\frac{dy}{dx}+2xy^3+2\frac{dy}{dx}=0

    Arrange with dy/dx terms on the left, all else on the right:

    3x^2y^2\frac{dy}{dx}+2\frac{dy}{dx}=-2xy^3

    Factor out dy/dx on the left side:

    \frac{dy}{dx}(3x^2y^2+2)=-2xy^3

    Divide through by 3x^2y^2+2

    \frac{dy}{dx}=-\frac{2xy^3}{3x^2y^2+2}
    Last edited by MarkFL; October 11th 2012 at 08:45 AM.
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  12. #12
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    Re: Implicit differentiation

    Just in case a picture helps (and otherwise to correct a small slip/typo, in Mark's differentiation of y^3)...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    But this is wrapped inside the legs-uncrossed version of...



    ... the product rule, where, again, straight continuous lines are differentiating downwards with respect to x.


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