What does the vertical bar mean in $ \left.\frac{\partial f}{\partial x}\right\rvert $

Question

I want to know what the symbol '|' besides a function means. For example: $$ \left.\frac{\partial f}{\partial x}\right\rvert $$

Answer 1

The partial derivative of $f$ with respect to $x$ evaluated at $x=2$. Written in $\LaTeX$ as \frac{\partial f}{\partial x}\Big{|}_{x=2} or alternatively use \right\rvert

\begin{align} \frac{\partial f}{\partial x}\Big{|}_{x=2} \end{align}

Answer 2

It is an old-fashioned piece of notation. The vertical bar is often used to mean that we must insert the conditions in the expression. For partial derivatives my humble opinion is that we should not use it regularly. The standard notation is $$ \frac{\partial f}{\partial x} (x_0,y_0), $$ like we do for a generic function of two variables in modern analysis.