How Can We Determine ∂z/∂r for a Complex Multivariable Equation?

[catch22]

Homework Statement

If z= x² - y² + xy
x = w + 4t
y = w² -5t + 4
w = r² - 5u
t = 3r + 5u

find ∂z/∂r

Homework Equations

The Attempt at a Solution

This is going to be quite long and tedious so bare with me, but here is my work.
drawing a diagram to help out a bit.

∂z/∂r = (∂z/∂x)(∂x/∂w)(∂w/∂r) + (∂z/∂x)(∂x/∂t)(∂t/∂r) + (∂z/∂y)(∂y/∂w)(∂w/∂r) + (∂z/∂y)(∂y/∂t)(∂t/∂r)

=(2x-y)(1)(2r) + (2x-y)(4)(3) + (-2y+x)(2w)(2r) + (-2y + x)(-5)(3)

now is that the final answer? or should everything be in terms of r?

 

[andrewkirk]

I haven't checked your final differentiations, but if they are correct then the answer will be equally correct regardless of the variables in terms of which it is expressed. It may however be more conventional to express it in terms of u and r, since they are the 'ultimate' parameters. It comes down to what you think your teacher will want. If you don't know then it would be safer to express it in terms of u and r.