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Math Help - First-Order Partial Derivatives

  1. #1
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    First-Order Partial Derivatives

    Calculate the first-order partial derivative of the following:



    for all (x,y) in R^2

    I used this as a composition function and used the chain rule.


    However, I am unsure of how to apply this formula.

    First-order partial derivative of x would be


    First-order pairtal derivative of y would be


    My question is : how to compute this? Do I do another composition+chain rule?

    Thank you.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Paperwings View Post
    Calculate the first-order partial derivative of the following:



    for all (x,y) in R^2

    I used this as a composition function and used the chain rule.


    However, I am unsure of how to apply this formula.

    First-order partial derivative of x would be


    First-order pairtal derivative of y would be


    My question is : how to compute this? Do I do another composition+chain rule?

    Thank you.
    You cannot, it doesn't tell you what the function is. You have gone as far as you can


    This is analgous to saying


    .

    Since in terms of the derivative in respect to x we don't care whether or not there are y's in there, this is just a normal chain rule of a function of x with some constants.
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    One more question: suppose that g has second derivative, how would I calculate the second partial derivatives of the function?
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  4. #4
    Moo
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    Hello,
    Quote Originally Posted by Paperwings View Post
    One more question: suppose that g has second derivative, how would I calculate the second partial derivatives of the function?
    It depends on what second partial derivative you want.

    First-Order Partial Derivatives-6.gif

    You'll get the first one by taking the partial derivative of df/dx with respect to y. You'll get the second one by taking the partial derivative of df/dx with respect to x.
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    My book doesn't specify, but I'm pretty sure I'm supposed to find and . Thank you, Moo.
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Paperwings View Post
    My book doesn't specify, but I'm pretty sure I'm supposed to find and . Thank you, Moo.
    Well then do what you did again

    for f_{xx} just hold y constant again and use either the chain rule and product rule or a combination of both if waranted. Same thing with [tex]f_{yy}[/math ] except now x is the variable held as a constant. if it is indeed f_{xy} note that if hte curve is continuous that f_{xy}=f_{yx} so you would only need to compute one.
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