multivariable chain rule

If $z=f(x,y)$ and $x,y$ both depend on $t$ , then $dz/dt$ sums both partials' contributions: $f_x\,dx/dt+f_y\,dy/dt$ .

$\dfrac{dz}{dt} = \dfrac{\partial z}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial z}{\partial y}\dfrac{dy}{dt}$ is a sum with one term per INPUT that itself depends on $t$ — $z$ feels $t$ through $x$ changing, and separately through $y$ changing, and the total rate is both contributions added together. This is the direct generalization of the ordinary chain rule, the special case where $z$ has only one input.

Examples

For $z=x^2y$ with $x=\cos t$ , $y=\sin t$ , at $t=\pi/2$ , $dz/dt=(2xy)(-\sin t)+(x^2)(\cos t)=0$ .