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As a symbol, I suppose I would call it a "bar".
To pronounce it in this context, I would say "evaluated at" and then whatever the subscript is. So $$ \frac{dy}{dx} \bigg |_{x_0}$$ might be pronounced "the derivative of $y$ with respect to $x$ evaluated at $x_0$". Others might say that $$ \frac{dy}{dx} \bigg |_{x_0} = \frac{dy}{dx}(x_0),$$ where the $(x_0)$ is the argument of the function $\frac{dy}{dx}$.
This is essentially the same sign that appears after evaluating integrals, if you have seen that before.
davidlowryduda
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