What does the long '|' mean? [duplicate]

Question

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• What is the name of the vertical bar? 4 answers

I have often seen the long '|' symbol with a subscript expression afterward.

What does this mean in mathematics? Here is an example I found from Wikipedia:

$$\large\left.\frac{dy}{dx}\right|_{x=c} \;\;= \;\;\;\;\left.\frac{dy}{du}\right|_{u=g(c)}\cdot \;\;\;\;\left.\frac{du}{dx}\right|_{x=c}$$

Answer 1

In this context, it means nothing more than to "evaluate the derivative at" the subscripted value. The $\Big|$ symbol is sometimes referred to as the "evaluation bar", with the "what to evaluate" preceding the symbol, and the "where to evaluate" to the right of the symbol.

So suppose $f(x) = x^2 - 1$. Then $\dfrac{dy}{dx} = 2x.$ Hence

$$\left.\frac{dy}{dx}\right|_{x = c} = \quad 2x\Big|_{x=c}\;\; =\quad 2c$$

Similarly, the evaluation bar is used in evaluating after integrating a definite integral, e.g. $$F(x)\Big|_a^b = F(b) - F(a)$$.

Answer 2

Here, it means "evaluate the derivative at".

With $y=x^2$, first one would read "$2x$ evaluated at $x=c$," which would yield $2c$.