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Functions of Several Variables - Differentiation Watch

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    Don't have any idea on how to approach this question. Anyone able to lend a hand?
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    (Original post by Sgt.Incontro)
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    Don't have any idea on how to approach this question. Anyone able to lend a hand?
    Just work out the partial derivatives of f; that's the method of approach. And show they satisfy those equations.
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    (Original post by ghostwalker)
    Just work out the partial derivatives of f; that's the method of approach. And show they satisfy those equations.
    1) How do you work out the partial derivatives?

    2) I think that df/dx is a partial derivative, so working out d^2f/dx^2 should be easy. But what about d^2f/dxdy for example? The first part looks trickier.

    Thanks for the help.
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    (Original post by Sgt.Incontro)
    1) How do you work out the partial derivatives?
    To work out a partial derivative wrt x say, you treat all the other variables as constants, and in effect differentiate as normal.

    2) I think that df/dx is a partial derivative, so working out d^2f/dx^2 should be easy. But what about d^2f/dxdy for example? The first part looks trickier.

    Thanks for the help.
    \dfrac{\partial^2f}{\partial x\partial y}= \dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)

    So, in this case, take the partial derivative wrt y first, and taking the result of that, take the partial derivative wrt x.
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    (Original post by ghostwalker)
    To work out a partial derivative wrt x say, you treat all the other variables as constants, and in effect differentiate as normal.



    \dfrac{\partial^2f}{\partial x\partial y}= \dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)

    So, in this case, take the partial derivative wrt y first, and taking the result of that, take the partial derivative wrt x.
    Makes a lot more sense now, thanks a lot!
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    (Original post by Sgt.Incontro)
    Makes a lot more sense now, thanks a lot!
    Good.

    You're welcome.
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    (Original post by ghostwalker)
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    So what would the partial derivatives be in this case? I.e df/dx and df/dy?

    Am I right in thinking that this is NOT like implicit differentiation?

    so df/dx = e^x(cos(y))
    df/dy = -e^x(sin(y)) ?
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    (Original post by Sgt.Incontro)
    So what would the partial derivatives be in this case? I.e df/dx and df/dy?

    Am I right in thinking that this is NOT like implicit differentiation?

    so df/dx = e^x(cos(y))
    df/dy = -e^x(sin(y)) ?
    Yes, it's not like implicit differentiation. More like standard differentiation.


    And those are correct.
 
 
 
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