多变量情形下的链式法则

Dependence flows along a tree

A worked chain, checked two ways

More variables, more paths

Notation discipline for composite derivatives

For $z = x^2 y$ with $x = t^2$ and $y = t^3$, what does the chain rule give for $\dfrac{dz}{dt}$ at $t = 1$?

If $w = f(x, y, z)$ with $x, y, z$ each functions of $t$, how many products does $\dfrac{dw}{dt}$ sum, and what does each product look like?

For $z = f(x, y)$ with $y = g(x, t)$, the variable $x$ influences $z$ in two ways. Which formula gives $\dfrac{dz}{dx}$ holding $t$ fixed?

For $z = xy$ with $x = t$ and $y = t^2$, compute $\dfrac{dz}{dt}$ at $t = 2$ and give the value, as a plain number.

Temperature is $T = f(x, y)$ with $f_x = 3$ and $f_y = 2$ at a point, and a probe moves with velocity $\left(\dfrac{dx}{dt}, \dfrac{dy}{dt}\right) = (1, 3)$ through that point. Give the rate $\dfrac{dT}{dt}$ the probe measures, as a plain number.