Using
$$ \left( \frac{\partial x}{\partial y} \right)_{z} \left( \frac{\partial y}{\partial z} \right)_{x} \left( \frac{\partial z}{\partial x} \right)_{y} $$
find $ \left( \frac{\partial x}{\partial t} \right)_{\rho_0} $ for the wave equation $y(x,t,\rho) = y_m \sin(\omega t-kx+\rho_0)$.
I have been staring at this problem for 20 minutes now, I have not come up with any way to attack it. My issue is that I need to find the partial derivative of $x$ with respect to $t$, when I don't have such a function. All I have is $y$, and all three variables are part of the sine argument.
From the wiki page:
$$dy=\left(\frac{\partial y}{\partial x}\right)_z dx+\left(\frac{\partial y}{\partial z}\right)_x dz$$ $$\frac{dy}{dx}=\left(\frac{\partial y}{\partial x}\right)_z +\left(\frac{\partial y}{\partial z}\right)_x \frac{dz}{dx}$$
In the same way: $$dz=\left(\frac{\partial z}{\partial x}\right)_y dx+\left(\frac{\partial z}{\partial y}\right)_x dy$$ $$\frac{dz}{dx}=\left(\frac{\partial z}{\partial x}\right)_y +\left(\frac{\partial z}{\partial y}\right)_x \frac{dy}{dx}$$
and if we combine the two:
$$\frac{dy}{dx}=\left(\frac{\partial y}{\partial x}\right)_z +\left(\frac{\partial y}{\partial z}\right)_x \cdot \left[\left(\frac{\partial z}{\partial x}\right)_y +\left(\frac{\partial z}{\partial y}\right)_x \frac{dy}{dx}\right]$$ $$\frac{dy}{dx}\cdot \left[1-\left(\frac{\partial y}{\partial z}\right)_x\cdot \left(\frac{\partial z}{\partial y}\right)_x\right]=\left(\frac{\partial y}{\partial x}\right)_z +\left(\frac{\partial y}{\partial z}\right)_x \cdot \left(\frac{\partial z}{\partial x}\right)_y $$ $$\frac{dy}{dx} = \frac{\left(\frac{\partial y}{\partial x}\right)_z +\left(\frac{\partial y}{\partial z}\right)_x \cdot \left(\frac{\partial z}{\partial x}\right)_y}{1-\left(\frac{\partial y}{\partial z}\right)_x\cdot \left(\frac{\partial z}{\partial y}\right)_x} $$
Now, if $y=y_m \sin(\omega t-kx+\rho _0)$, let us write: $R(y,x,t)=y_m \sin(\omega t-kx+\rho _0)-y=0$ . Now from this link we can write: $$\frac{dy}{dx}=-\frac{\frac{\partial R}{\partial x}}{\frac{\partial R}{\partial y}}$$
I will not finish the job. I will refer you to this wiki page, so you can finish the job yourself.