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Thread: Partial Derivatives

  1. #1
    Senior Member Vinod's Avatar
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    Partial Derivatives

    Hello Forumites,

    The partial derivatives of order r of an analytic function $f(x_1,...x_n)$ of n variables do not depend on the order of differentiation but only on the number of times that each variables appears.And hence there exists $\binom{n+r-1}{r}$ different partial derivatives of rth order. A function of three variables has fifteen derivatives of fourth order and 21 derivatives of fifth order.
    What does this means? If any one knows, he may reply.
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    Re: Partial Derivatives

    There are some "she's" on this forum too … just sayin'
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  3. #3
    Senior Member Vinod's Avatar
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    Re: Partial Derivatives

    Quote Originally Posted by Debsta View Post
    There are some "she's" on this forum too just sayin'
    Hello,
    If any forumite knows the answer, post it into this thread. Suppose $f(x,y,z)=3x^4+3y^2-3z^2-56$. Now in this function there are three variables,how many partial derivatives can be formed?
    Last edited by Vinod; Sep 23rd 2018 at 04:44 AM.
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    Re: Partial Derivatives

    "A function of three variables has fifteen derivatives of fourth order."

    Lets say your variables are x, y and z.

    For the fourth order partial derivatives, you need to look at what you can take these derivatives with respect to. Order makes no difference.

    So you could take the fourth derivative wrt x then x then x then x again - for simplicity (and so I don't have to write in LaTex) let's call that xxxx.

    So basically you are finding how many strings of 4 you can make using the variables x, y and/or z.

    xxxx
    xxxy
    xxyy
    xyyy

    yyyy
    yyyz
    yyzz
    yzzz

    zzzz
    zzzx
    zzxx
    zxxx

    xyzz
    xyyz
    xxyz .... 15 of them! n+r-1 = 3+4-1=6 ; r=4 ; 6C4 =15
    Thanks from Vinod
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    Re: Partial Derivatives

    I feel like it's important to point out that mixed partial derivatives are not necessarily equal. Here is an example: https://en.wikipedia.org/wiki/Symmet..._of_continuity.

    Since you said the function is analytic, though, it is infinitely differentiable. And the polynomial you give is infinitely differentiable. Because of this, Debsta's answer is correct.

    The number of ways to choose r things from a group of n (with replacement, order does not matter) is the formula you gave: $\displaystyle \binom{n+r-1}{r}$.

    - Hollywood
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