Holding things constant in partial derivatives

Question

Suppose we have x(u,v), w(x,y), and y(x). If I took the partial derivative of w with respect to u while keeping x constant then, $$\left(\frac{\partial w}{\partial u}\right)_x=\frac{\partial w}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial x}\frac{\partial x}{\partial u}$$ so is it true that this is simply zero? Its probably a bad example but essentially if I kept x constant then is $\partial x$ automatically zero? I was confused because I was thinking "since $\frac{\partial x}{\partial u}$ means the change in x because of a little change in u while keeping v constant, and since v and u arent necessarily constants but somehow balence eachother to make x a constant. So the change in x over the change in u isn't constant??" what did i get wrong?

Answer 1

v and u arent necessarily constants but somehow balence eachother to make x a constant

It's true that $v$ would change as $u$ changes in order to keep $x$ constant. But that means $\left(\dfrac{\partial v}{\partial u}\right)_x$ is not typically $0$.

But that says nothing about $\left(\dfrac{\partial x}{\partial u}\right)_x$ . Indeed, if $x$ is constantly $17$, then $x$ is not allowed to change as $u$ changes, so the change in $x$ would be $0$ if we let it be defined in the first place.

As a result, if $x$ is constantly $x_0$, then $w$ is constantly $w(x_0,y(x_0))$ and doesn't change either.