Partial Derivative Manipulation for Physical Chemistry Homework problem

[Jayjayjay]

Homework Statement

Given the functions Q(v,w) and R(v,w)
[/B]
K = v(dQ/dv)r and L = v(dQ/dv)w

Show that
(1/v)K = (1/v)L + (dQ/dw)v (dW/dv)r

I have the problem attached if for clarity of the information.

Homework Equations

I assume everything is given in the problem.

The Attempt at a Solution

Q(v,w) --> dQ = (dQ/dv)w DV + (dQ/dw)v DW
R(v,w) --> dR = (dR/dv)w DV + (dR/dw)v DW

(1/v)K = (dQ/dv)r (Equation 1)
so I have to manipulate the given information to equal (dQ/dv)r and substitute (I think)

(1/v)L = (dQ/dv)w
which is in the function Q(v,w) so I was thinking you could just make

dQ - (dQ/dw)v DW = (dQ/dv)w

and substitute to equal

(1/v)L = dQ - (dQ/dw)v dW --> (1/v)L + (dQ/dw)v dW = dQ

If I divided the whole thing by dv I could get

(1/v)L + (dQ/dw)v (dW/dv)r = (dQ/dv)r

and substitute this into equation 1 to get

(1/v)K = (1/v)L + (dQ/dw)v (dW/dv)r

Is this correct? I'm not sure. [/B]

 

[Chestermiller]

If R=R(v,w), what is dR in terms of dv and dw (and the partial derivatives of R with respect to v and w)? How do dv and dw have to vary if dR = 0? What is the partial derivative of w with respect to v at constant R?

Chet