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

• Oct 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.
• Oct 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.
• Oct 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.