solve dI/dx for I = (hx + hz - wz)/(xy + xz + yz)?

Answer 1

The answer depends on a few things:
- did you mean dI/dx or ∂I/∂x?
- are h, x, y, z, w independent, or constrained?

If you mean ∂I/∂x, OR if h, x, y, z and w are all independent, then the answer is just what it would be if h, y, z and w were all constants:
∂I/∂x = [h(xy + xz + yz) - (hx + hz - wz)(y + z)] / [(xy + xz + yz)^2]
= [hxy + hxz + hyz - (hxy + hxz + hyz + hz^2 - wyz - wz^2)] / [(xy + xz + yz)^2]
= [z(wy + wz - hz)] / [(xy + xz + yz)^2]

If you really want dI/dx and h, w, x, y and z are dependent, then you have to calculate it with the total derivative formula:

dI/dx = ∂I/∂x + ∂I/∂h ∂h/∂x + ∂I/∂w ∂w/∂x + ∂I/∂y ∂y/∂x + ∂I/∂z ∂z/∂x

which is not going to be terribly nice to compute unless ∂h/∂x and so on help you out a bit.