Question on Proof of a Wave Speed on a Stretched String

Question

I do not understand why the partial derivative of y with respect to x is

$\frac{F_{1}}{F_{t}} = \frac{-dy}{dx}$

Shouldn't the partial derivative of y w.r.t x = 0?

Answer 1

I believe that the explanation is given in the figure caption. In particular, the slope of the string is $\frac{\partial{y}}{\partial{x}}$ which is equal to $\tan {\theta}$ as implied in the figure. More generally, $y$ will be a function of both $x$ and $t$, where $t$ is the time, but the analysis above is looking at the situation at a frozen instant of time, then considering how $y$ varies with position $x$ based on the forces as specified. However, since $y$ is a function of both $x$ and $t$, one needs to use partial derivatives to be consistent.

I hope this helps.