I'm given the following two relationships between $x, y, z$: $$\begin{cases} f(x,y,z)=x+y+z-1=0 \\ g(x,y,z)=x^2-2y^2+3z^2-2=0\end{cases} \quad .$$
Question:
How would I go about calculating $\frac{\mathrm{d}z}{\mathrm{d}x}$ and $\frac{\mathrm{d}y}{\mathrm{d}x}$ (in terms of $x,y,z$)?
$$\underline{\text{My thoughts}}$$ From the chain rule, I can differentiate (and then divide through by $2$) to get the following two relationships between the differentials: $$\begin{cases} \mathrm{d}x+\mathrm{d}y+\mathrm{d}z=0 \\ x\mathrm{d}x-2y\mathrm{d}y+3z\mathrm{d}z=0\end{cases} \quad ,$$ but where would I go from here? Is there something simple that I'm missing?
You could take the $x$-derivative of both equations to find \begin{align*} & 1 + \frac{\mathrm{d} y}{\mathrm{d}x} + \frac{\mathrm{d} z}{\mathrm{d} x} = 0 \\ & 2x - 4y \frac{\mathrm{d} y}{\mathrm{d}x} + 6z \frac{\mathrm{d} z}{\mathrm{d}x} = 0 \end{align*}
Now you have two equations for two unknowns. Solving this system for $\frac{\mathrm{d} y}{\mathrm{d}x}$ and $\frac{\mathrm{d} z}{\mathrm{d}x}$ should be a piece of cake.