Interpretation of partial derivatives of vertical coordinate with respect to $x$ and time

Question

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.

Thanks,

Answer 1

I think it should say $y_t(x,y)$ is the rate of change of the function along the $t$-axis (not the $x$-axis). Whichever variable you differentiate with respect to, everything else remains constant.

The geometric intuition I think is usually the best for illustrating what derivatives are. Think of it as the definition of an ordinary derivative. You give the argument a little change and look at what happens with the function value. If that change is infinitely small, then the ratio of the change in the function value and the argument change is your derivative. Now think of some surface $z=z(x,y)$ that you want to find partial derivatives of. If you take a slice of that surface parallel to the $xz$-plane, the surface will leave a path on that slice. If you look at that path as a single argument function (i.e. a slice of that surface with a constant $y$-value), your definition of a partial derivative w.r.t $x$ reduces to an ordinary derivative. If you want to differentiate w.r.t $y$, slice parallel to the $yz$-plane.

Answer 2

This requires some physics intuition, but the rate of change with respect to $t$ of a position (like $y(x,t)$ in this case) is a velocity.

Movement in the $y$ direction is vertical, so in this case $y_t(x,t)$ describes the vertical velocity at a time $t$ and a position $x$.