Number of terms for expression using the chain rule

Question

I am given the following:

$z = g(u, v)$

$u = u(x, y, t)$

$v = v(x, y, t)$

$x = x(t)$

$y = y(t)$

And I need to find the number of terms for $dz/dt$ (using the chain rule)

Drawing out a tree I have...

        z
    /       \
   u         v
 / | \      / | \
x  y  t    x  y  t
|  |       |  |
t  t       t  t

Applying the chain rule, I believe the terms would be...

${dz\over{du}} \times {du\over{dx}} \times {dx\over{dt}} + {dz\over{du}} \times {du\over{dy}} \times {dy\over{dt}} + {dz\over{du}} \times {du\over{dt}} + {dz\over{dv}} \times {dv\over{dx}} \times {dx\over{dt}} + {dz\over{dv}} \times {dv\over{dy}} \times {dy\over{dt}} + {dz\over{dv}} \times {dv\over{dt}}$

Are there 6 terms here?

Answer 1

Yes, this is correct (now, after the last edit): by the Chain Rule, there are indeed $6$ terms there. And your expression is pretty much correct too, except for inaccurate notation: for functions of several variables, their derivatives in this expression must be partial derivatives. So it must look as: $${\partial z\over{\partial u}} \times {\partial u\over{\partial x}} \times {dx\over{dt}} + {\partial z\over{\partial u}} \times {\partial u\over{\partial y}} \times {dy\over{dt}} + {\partial z\over{\partial u}} \times {\partial u\over{\partial t}} + {\partial z\over{\partial v}} \times {\partial v\over{\partial x}} \times {dx\over{dt}} + {\partial z\over{\partial v}} \times {\partial v\over{\partial y}} \times {dy\over{dt}} + {\partial z\over{\partial v}} \times {\partial v\over{\partial t}}$$