My question is from my lecturers notes, this is what he wrote and I don't know what he is on about :
What is a physical meaning of partial derivatives of $y(x,t)$? $y_x(x,t)$ is the rate of change of the function along the $x$-axis,i.e. a slope of the string at a point x at a given instant of time.
So just hold $t$ constant and what is the rate of change of $x$ at a constant $t$?
Then he says:
$y_t(x,t)$ is the rate of change of the function along the $x$-axis i.e. a vertical velocity of point on the string, having a horizontal coordinate $x.$
I have no idea about that. I would think that the second one would just mean the rate of change of $t$ when $x$ is constant but plainly I don't understand something here.
I guess it could also mean if the overall function is in terms of $t$ and $x$ then if $t$ changes so does the rate of change of $x$ for a given value of $t.$ Because it is like you're taking lots and lots of different slices of some object. That is the only thing I can think of.
Sorry, not looking for the mathematical definitions just trying to understand this intuitively.