What is the difference between $y_{|x=1}=2$ and $y(1)=2$ What is the difference between $$y_{|x=1}=2$$ and $$y(1)=2$$
Let's say you have an equation
$$xy^3+y^2-y+2=C$$ and the value of $C$ is asked when $y_{|x=1}=2$. Can I just directly input the values?
 A: Yes. Here, $y_{|x=1}=2$ and $y(1)=2$ are the same. Indeed, this notation is widely used.
General definition. For a mapping $f:A\to B$ and a subset $C\subset A$, we use $f|_C$ to denote the mapping $g: C\to B$ given by $g(c)=f(c)$ for all $c\in C$, and the mapping $f|_C$ is called the restriction of $f$ to $C$.
Here, we have a function $y:\mathbb R\to\mathbb R$ and $\{1\}\subset \mathbb R$ is a subset, and then $y|_{x=1}$ is the same as $y|_{\{x=1\}}$, or $y|_{\{1\}}$ defined above. We often use $y|_{x=1}$to emphasize that the independent variable is denoted by $x$.
If you've known $xy^3+y^2-y+2=C$ and $y|_{x=1}=2$ and you want to know the value of $C$, then you just need to let $x=1$ and $y=2$ in $xy^3+y^2-y+2=C$ to get $$C=1\cdot2^3+2^2-2+2=12.$$
A: Let $f:\mathbb R^2 \to \mathbb R,(x,y)\mapsto xy^3+y^2-y+2$.
$f_y^{'}(1,2)\neq 0$. As already explained, you can even write  :$$f_y^{'}(1,2)=[3xy^2+2y-1]_{/(1,2)}=15$$
and $f_x^{'}(1,2)=[y^3]_{/(1,2)}=8$ and $[f(x,y)]_{/(1,2)}=12$ (the last result by hypothesis.)
Since $f_y^{'}(1,2)\neq 0$, the equation $f(x,y)=12$ defines an implicit function $\varphi$, that by abuse of notation, we will continue to note y here, as has already been explained.
You know that $\varphi'(1)=-\frac{8}{15}$...
A: If $y=3x^2-1$, note that
$$
(3x^2-1)_{|x=1}=2
$$
should not be written
$$
(3x^2-1)(1)=2
$$
