What's the partial derivatives (dx/du) of x=u/u+v?

Answer 1

Though you say derivatives (plural), I presume that you only want the partial derivative (singular) that you indicate in writing.

If u and v are some other independent variables, the first thing is to simplify the RHS for the differentiation wrt u, by writing it as:

x = u/(u + v) = 1 - v/(u + v).

Now, there's only one 'u' to concern oneself with.

Clearly, then, (partial) del x/del u = +v/(u + v)^2. QED

Of course, (partia) del x/del v = -u/(u + v)^2, from the original form of the RHS, just in case you were interested.

Live long and prosper.

P.S. It's been > 50 years for me!

Answer 2

If I remember correctly (it's been 15 years), you treat the v as a constant, so by the quotient rule:

dx/du = [(u+v)(1) - u(1)]/(u+v)² = v/(u+v)²

I think?

Answer 3

As written, 0. If you meant x = u/(u+v), the standard calculus rules will tell you easily enough; I am feeling lazy this evening and will let you deal with it.

Answer 4

x=u/(u+v)
dx/du = ((u+v) - u)/(u+v)^2
dx/du = (u+v-u) / (u+v)^2
dx/du = v / (u+v)^2