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Math Help - Total Differentiation with Nested, Unspecified Composite Functions

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    Newbie jrhorn424's Avatar
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    Question Total Differentiation with Nested, Unspecified Composite Functions

    I have no idea if the thread title accurately explains the question I'm having, but it is the best I could do. Googled and searched the forum and couldn't find what I was looking for.

    Basically, I'm familiar with total differentiation. For instance I'm fairly certain that for a function

    U=u(C(R),E(R)),

    the total derivative is

    {dU \over dR} = U_CC_R + U_EE_R.

    But what if the composite functions are functions of many variables. For instance,

    U=u(C(Y,R_D),E(R_D,R_F)).

    My first attempt for the total differential w.r.t. R_D:

    {dU \over dR} = U_CC_Y + U_CC_{R_D}+U_EE_{R_D}+U_EE_{R_F}

    My second attempt:

    {dU \over dR} = U_CC_YY_{R_D} + U_CC_{R_D}+U_EE_{R_D}+U_EE_{R_F}{R_F}_{R_D}

    Are either correct? Any help is appreciated!
    Last edited by jrhorn424; December 3rd 2009 at 12:47 PM. Reason: Added the last derivative.
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