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Math Help - Second Order Partial Derivatives

  1. #1
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    Second Order Partial Derivatives

    Just need pointing in the right direction then hopefullly I'l understand these ....

    Find 1st and 2nd Order Partial Derivatives:

    z = 3x - 2y⁴

    First Order
    = 2x
    = -8y

    Now I know there should be 4 second order Derivatives and i remember how to work the first two out which are:

    = 2
    = -24y

    However I become stuck on working out the last two which I believe should be the same.

    I know what they are they are the Zxy and Zyx and I know they are the same but could someone explain how i calculate these thanks.

    �I know what theI kk
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  2. #2
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    Quote Originally Posted by DwightHoward View Post
    Just need pointing in the right direction then hopefullly I'l understand these ....

    Find 1st and 2nd Order Partial Derivatives:

    z = 3x - 2y⁴

    First Order
    = 2x
    = -8y
    Better to write explicitely that z_x=6x (NOT 2x) and z_y= -8y^3.
    z_{xx}= (z_x)_x= 6 (NOT 2) and z_{yy}= (z_y)_y= -24y^2/

    z_{xy}= (z_x)_y. What is the derivative of 6x with respect to y?
    z_{yx}= (z_y)_x. What is the derivative of [itex]-8y^3[itex] with respect to x?

    Now I know there should be 4 second order Derivatives and i remember how to work the first two out which are:

    = 2
    = -24y

    However I become stuck on working out the last two which I believe should be the same.

    I know what they are they are the Zxy and Zyx and I know they are the same but could someone explain how i calculate these thanks.

    �I know what theI kk
    Follow Math Help Forum on Facebook and Google+

  3. #3
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    thanxxx
    Last edited by DwightHoward; July 2nd 2010 at 04:47 AM.
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  4. #4
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    Quote Originally Posted by DwightHoward View Post
    Firstly just realised myself i forgot to times by 3 but thats no problem.

    Secondly am I rite to assume 6x cannot be differentiated with respect to y and is therefore 0?

    If not would you please explain how it is calculated

    Thanks for your help
    Yes, your answer is correct. However, the correct wording would be to say that 6x does not depend on y, and therefore z_{xy}=0. Just for clarification: 6x can be differentiated with respect to y, but that differentiation gives a result of 0.
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