[SOLVED] chain rule with functions related by equations?

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October 19th 2009, 03:43 PM
kodos

[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

$1) x+y= uv$
$ 2) xy=u-v $

and

$ x=g(u,v)$
$y=h(u,v) $

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

$ g(u,v)=uv-y $
$ 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.
October 20th 2009, 12:20 PM
kodos

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.
October 21st 2009, 07:51 AM
kodos

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.