How to find the derivative of a multivariable function with respect to one of its variables

Question

If I have a function $g(x,y)$, what is the formula for calculating $\frac{dg}{dx}$? I know that if $x,y$ are functions of $t$, then $\frac{dg}{dt} = \frac{\partial g}{\partial x}\frac{dx}{dt}+\frac{\partial g}{\partial y}\frac{dy}{dt}$. Is it the same thing for differentiating with respect to $x$, and just replacing $t$ with $x$ in the equation?

Answer 1

Yes, it is exactly so.

$\frac{dg}{dx}=\frac{\partial g}{\partial x}\frac{dx}{dx}+\frac{\partial g}{\partial y}\frac{dy}{dx}$

It has nothing to do with the naming of variables. This formula is due to the fact that maybe $y$ is dependent on $x$, as in putting $y=x^3$.