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Math Help - Partial Derivative of Higher Orders

  1. #1
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    Partial Derivative of Higher Orders

    Hello, everyone. It's been awhile since I've visited the boards. I hope all is well for you and yours.

    The problem I'm dealing with is as follows:

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

    z = u{(v-w)}^{\frac{1}{2}}

    Find the indicated partial derivative:

    \frac{\partial^3z}{{\partial}u{\partial}v{\partial  }w}

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

    I'm just wondering if anyone can help explain the problem to me as I'm having trouble getting started. I don't really know where to find a thread. Any tips or hints would be appreciated. I guess the partial derivative it's asking for is the third parital derivative with respect to u, v, and w?

    Thanks in advance for your consideration,

    -Austin
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by auslmar View Post
    Hello, everyone. It's been awhile since I've visited the boards. I hope all is well for you and yours.

    The problem I'm dealing with is as follows:

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

    z = u{(v-w)}^{\frac{1}{2}}

    Find the indicated partial derivative:

    \frac{\partial^3z}{\partial u \partial v\partial w}

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

    I'm just wondering if anyone can help explain the problem to me as I'm having trouble getting started. I don't really know where to find a thread. Any tips or hints would be appreciated. I guess the partial derivative it's asking for is the third parital derivative with respect to u, v, and w?

    Thanks in advance for your consideration,

    -Austin
    Note that \frac{\partial^3z}{\partial u\partial v\partial w}=\frac{\partial}{\partial u}\left(\frac{\partial^2z}{\partial v\partial w}\right)=\frac{\partial}{\partial u}\left[\frac{\partial}{\partial v}\left(\frac{\partial z}{\partial w}\right)\right]

    So first [partially] differentiate z with respect to w, then [partially] differentiate that result with respect to v, and finally [partially] differentiate that result with respect to u.

    Does that make sense?

    --Chris
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    Note that \frac{\partial^3z}{\partial u\partial v\partial w}=\frac{\partial}{\partial u}\left(\frac{\partial^2z}{\partial v\partial w}\right)=\frac{\partial}{\partial u}\left[\frac{\partial}{\partial v}\left(\frac{\partial z}{\partial w}\right)\right]

    So first [partially] differentiate z with respect to w, then [partially] differentiate that result with respect to v, and finally [partially] differentiate that result with respect to u.

    Does that make sense?

    --Chris
    It makes sense to me. Tell me if I've gone wrong.

    This is what I did with your suggestion:

    [(\frac{-u}{2{{\sqrt{{v-w}}}}})(\frac{u}{2{\sqrt{{v-w}}}})]\sqrt{{u-v}} = \frac{-u^2}{4{\sqrt{{v-w}}}}

    Is this the right idea? Thanks for your help.
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by auslmar View Post
    Hello, everyone. It's been awhile since I've visited the boards. I hope all is well for you and yours.

    The problem I'm dealing with is as follows:

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

    z = u{(v-w)}^{\frac{1}{2}}

    Find the indicated partial derivative:

    \frac{\partial^3z}{{\partial}u{\partial}v{\partial  }w}

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

    I'm just wondering if anyone can help explain the problem to me as I'm having trouble getting started. I don't really know where to find a thread. Any tips or hints would be appreciated. I guess the partial derivative it's asking for is the third parital derivative with respect to u, v, and w?

    Thanks in advance for your consideration,

    -Austin
    Quote Originally Posted by auslmar View Post
    It makes sense to me. Tell me if I've gone wrong.

    This is what I did with your suggestion:

    [(\frac{-u}{2{{\sqrt{{v-w}}}}})(\frac{u}{2{\sqrt{{v-w}}}})]\sqrt{{u-v}} = \frac{-u^2}{4{\sqrt{{v-w}}}}

    Is this the right idea? Thanks for your help.
    z = u{(v-w)}^{\frac{1}{2}}

    Thus, \frac{\partial z}{\partial w}=-\frac{u}{2(v-w)^{\frac{1}{2}}}

    Now, \frac{\partial}{\partial v}\left[\frac{\partial z}{\partial w}\right]=\frac{\partial^2z}{\partial v \partial w}=\frac{u}{4(v-w)^{\frac{3}{2}}}

    Finally, \frac{\partial}{\partial u}\left[\frac{\partial}{\partial v}\left[\frac{\partial z}{\partial w}\right]\right]=\frac{\partial^3z}{\partial u\partial v \partial w}=\color{red}\boxed{\frac{1}{4(v-w)^{\frac{3}{2}}}}

    Does this make sense?

    --Chris
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  5. #5
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    Oh yeah, I see now. I was trying to multiply them all together. Thanks for your help.
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