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Math Help - [SOLVED] chain rule with functions related by equations?

  1. #1
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    [SOLVED] chain rule with functions related by equations?

    I just don't get this problem, I feel I'm missing something obvious.

    Find dg/du if

    1) x+y= uv
     <br />
2) xy=u-v<br />

    and

     <br />
x=g(u,v)
    y=h(u,v)<br />

    Poor attempt of solution:

    I try to obtain an expression for x as a function of u and v, but I can't seem to be able to get anything from that system of equations.

    From 1) I get x= uv-y , but from 2) y=(u-v)/x

    If I replace into 1) x=uv-uv/x

    I don't think I'm getting something from that

    If x=g(u,v) then

    <br />
g(u,v)=uv-y <br />
    <br />
g(u,v)=uv-h(u,v)

    Another thought from there: dg/du=v-dh/du

    h(u,v)=y=(u-v)/x

    dh/du=x

    then,

    dg/du=v-x , but x=g(u,v)

    and I think I'm running in circles here.
    Last edited by kodos; October 20th 2009 at 01:21 PM.
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  2. #2
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    What if I tried to obtain a F(x,y)= 0 which I could then derive and get an expression for dg/du? The problem would be getting rid of u and v, but the truth is I suck at solving systems of equations. Maybe adding and multiplying the equations...

    Another idea would be that y could be defined implicitly as a function of x.
    It would be like F(x,y)=F(x,f(x))=0, where x is defined by u and v and keeps holding the condition.
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  3. #3
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    I finally got the resolution for this exercise. The idea was derive both equations in terms of u. Then, having a system of equations, solve for dg/du.
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