What is the physical meaning when we do partial differential to a variable of equation?take this for example:
$y=ax+bz+c$, $x$ and $z$ are both variables in function y
now we do do partial differential to $x$,that is ,$\frac{\partial y}{\partial x}$,then what is the physical meaning of $\frac{\partial y}{\partial x}$? Does it mean that i want to know the different of $x$ in the function y ?
The partial derivative of $f$ in a certain direction gives you the rate of change of $f$ in that direction. So in your example $\frac{\partial y}{\partial x} = a$. This means that moving along the positive x-direction by one unit will increase the value of $y(x,z)$ by an amount $a$ (or decrease if $a$ is negative). This does not tell you about the rate of change of $y$ in any other direction. In particular, it tells you nothing about the rate of change of $y$ in the $z$ direction, i.e. you do not know how $y$ would change if you were to move up and down the $z$ axis. you also do not know how $y$ would change if you were moving along the line $x=z$, or the parabola $x= z^2$ etc. etc.
