We know that Chain rule of partial derivatives is something just like this ($z$ is function of $x$ and $y$ variable and, $x$ and $y$ is function of $t$) :
$$\frac{dz}{dt}=\frac{\partial z}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial z}{\partial y} \frac{\partial y}{\partial t}$$
Then, I was thinking to getting backward from the equation.
$$\frac{\partial z}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial z}{\partial y} \frac{\partial y}{\partial t} =\frac{\partial z}{\partial t}+\frac{\partial z}{\partial t}=2\frac{\partial z}{\partial t}$$
Can't we write that $\frac{\partial z}{\partial t}=\frac{dz}{dt}$? If not, than what's the difference between them? If yes, than why I couldn't prove backward?
From my study of Calculus, I read that $\frac{d}{dt}$ is used for total differentiation. And, $\frac{\partial }{\partial t}$ is some differentiation under $\frac{d}{dt}$. But, I am not sure if that's correct.
