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Math Help - Beginner Implicit differentiation

  1. #1
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    Beginner Implicit differentiation

    Here's the problem:
    x^4+x^2y+8y^2=10, Find y' by implicit differentiation.

    I set it up like this by taking the derrivative of both sides:
    \frac{\mathrm{d} y}{\mathrm{d} x}[x^4]+\frac{\mathrm{d} y}{\mathrm{d} x}[x^2y]+\frac{\mathrm{d} y}{\mathrm{d} x}[8y^2]=\frac{\mathrm{d} y}{\mathrm{d} x}[10]

    Giving me:
    4x^3+2xy+x^2y'+16yy'=0

    then:
    x^2y'+16yy'=4x^3+2xy

    Factor out y':
    y'(x^2+16y)=4x^3+2xy

    divide and the answer:
    y'=\frac{4x^3+2xy}{x^2+16y}

    This answer is wrong though. Can someone help me see where I went wrong? I'm wondering if {\frac{\mathrm{d} y}{\mathrm{d} x}}{[x^2y]}\neq (2xy+x^2y')

    Thanks in advance!
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  2. #2
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    Re: Beginner Implicit differentiation

    Quote Originally Posted by jjtjp View Post
    Here's the problem:
    x^4+x^2y+8y^2=10, Find y' by implicit differentiation.

    I set it up like this by taking the derrivative of both sides:
    \frac{\mathrm{d} y}{\mathrm{d} x}[x^4]+\frac{\mathrm{d} y}{\mathrm{d} x}[x^2y]+\frac{\mathrm{d} y}{\mathrm{d} x}[8y^2]=\frac{\mathrm{d} y}{\mathrm{d} x}[10]

    Giving me:
    4x^3+2xy+x^2y'+16yy'=0

    then:
    x^2y'+16yy'=4x^3+2xy This should be x^2y'+16yy'=-4x^3-2xy

    Factor out y':
    y'(x^2+16y)=4x^3+2xy

    divide and the answer:
    y'=\frac{4x^3+2xy}{x^2+16y}

    This answer is wrong though. Can someone help me see where I went wrong? I'm wondering if {\frac{\mathrm{d} y}{\mathrm{d} x}}{[x^2y]}\neq (2xy+x^2y')

    Thanks in advance!
    See above.
    Thanks from jjtjp
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  3. #3
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    Re: Beginner Implicit differentiation

    Wow..I solved the problem 3 times and missed it every time. How the heck?! Anyways, thank you!
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  4. #4
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    Re: Beginner Implicit differentiation

    Quote Originally Posted by jjtjp View Post
    Here's the problem:
    x^4+x^2y+8y^2=10, Find y' by implicit differentiation.

    I set it up like this by taking the derrivative of both sides:
    \frac{\mathrm{d} y}{\mathrm{d} x}[x^4]+\frac{\mathrm{d} y}{\mathrm{d} x}[x^2y]+\frac{\mathrm{d} y}{\mathrm{d} x}[8y^2]=\frac{\mathrm{d} y}{\mathrm{d} x}[10]
    Do NOT write " \frac{\mathrm{d} y}{\mathrm{d} x}[x^4]", the "y" should not be there. You mean \frac{\mathrm{d} x^4}{\mathrm{d} x} or \frac{\mathrm{d} }{\mathrm{d} x}[x^4]

    Giving me:
    4x^3+2xy+x^2y'+16yy'=0

    then: x^2y'+16yy'=4x^3+2xy
    NO, you have subtracted 4x^3+ 2xy from both sides. This should be x^2y'+ 16yy'= -(4x^3+ 2xy) or x^2y'+ 16yy'= -4x^3- 2xy

    Factor out y':
    y'(x^2+16y)=4x^3+2xy

    divide and the answer:
    y'=\frac{4x^3+2xy}{x^2+16y}

    This answer is wrong though. Can someone help me see where I went wrong? I'm wondering if {\frac{\mathrm{d} y}{\mathrm{d} x}}{[x^2y]}\neq (2xy+x^2y')

    Thanks in advance!
    Last edited by HallsofIvy; March 8th 2013 at 01:24 PM.
    Thanks from jjtjp
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  5. #5
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    Re: Beginner Implicit differentiation

    Thanks for the correction.
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