If $\frac d{dx} f(x,y)>0$, can I claim that $f(x,y)$ is increasing with respect to $x$?

Question

I have an implicit equation $f(x,y)=0$; computing the derivatives, I see that $\frac d{dx} f(x,y)>0$ while $\frac d{dy} f(x,y)$ maybe positive, or negative.

Question. Is this data sufficient to claim that $f(x,y)$ is increasing with respect to $x$?

Answer 1

If you want to talk about whether it is increasing with respect to $x$, you need to let $y$ be given and fixed and then you look at the partial derivative with respect to $x$ for this fixed $y$. In your case, since you have computed the derivative and it is positive, you conclude that $f$ is strictly increasing with respect to $x$ for every fixed $y$.