Applying the chain rule using trees?

[Pizzerer]

Homework Statement

Write out a tree (this will be a big tree) of dependencies and hence write down an expression for
∂z/∂r

Homework Equations

z=k(x, y)=xy², x=(w₁)(w₂)+w₃, y=w₄;

w₁=t, w₂=t², w₃=2t+1, w₄=sin(t);
t=r²+2s²

The Attempt at a Solution

This is the tree I drew and followed the relevant paths in order to try and write and expression but my expression is to short. I only calculate 2 terms when there is suppose to be 8:
http://www.picpaste.com/Untitled_Image-OZCiTMvb.jpg

This is wrong however. I need 16 leaves at the base. What am I doing wrong?

 

[Mark44]

Homework Statement

Write out a tree (this will be a big tree) of dependencies and hence write down an expression for
∂z/∂r

Homework Equations

z=k(x, y)=xy², x=(w₁)(w₂)+w₃, y=w₄;

w₁=t, w₂=t², w₃=2t+1, w₄=sin(t);
t=r²+2s²

The Attempt at a Solution

This is the tree I drew and followed the relevant paths in order to try and write and expression but my expression is to short. I only calculate 2 terms when there is suppose to be 8:
http://www.picpaste.com/Untitled_Image-OZCiTMvb.jpg

This is wrong however. I need 16 leaves at the base. What am I doing wrong?

I drew a diagram and got the same one you did.

As far as your partial $\partial z/\partial r$ is concerned, I get four terms to your two. Three of the four terms look like this:
$$ \frac{\partial z}{\partial x} \frac{\partial x}{\partial w_i} \frac{d w_i}{dt} \frac{\partial t}{\partial r}$$

For the fourth one, the first factor is the partial of z with respect to y.

 

[SammyS]