Can the differentiation variable be simplified to just one variable?

[womfalcs3]

I'm working out some problems, and I'm ending up with a term similar to the following:

du/d(y/u)

I'm differentiating with respect to y/u. Both y and u are variables. How can I divide that up to represent differentiation with just one variable (Even if it means expanding the term)?

Is it mathematically possible to do that?

Thanks.

 

[junglebeast]

I'm really sure what you're trying to do, but if you have a function f(x,y) and you want to know how f changes in the limiting case with respect to a change in the ratio of y/x, then this is a case for the directional derivative. Note that v = (x, y) is the vector where y/x stays constant, so you'd want to take the vector orthogonal to that; ie, use v = (y, -x). In that case, \nabla_v f(x,y) = \nabla f(x,y) \cdot v

 

[HallsofIvy]

Let v= u/y. Then, if f(u,y) is any function of u and y, df/dv= \partial f/\partial u \partial u\partial v+ \partial f/\partial y \partial y/partial v.<br /> <br /> Since, here, v= u/y, so u= yv and \partial u/\partial v= y. Similarly, y= u/v so \partial y/\partial v= -u/v^2= -u/(u^2/y^2)= -y^2/u.<br /> <br /> That is, df/dv= y\partial f/\partial u- (y^2/u)\partial f/\partial y<br /> <br /> And, since here f(u,y)= u, \partial f/\partial u= 1 and \partial f/\partial y= 0 so we have df/dv= y.