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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:

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

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marked as duplicate by MJD, Amzoti, Micah, Joe, Dominic Michaelis Apr 10 '13 at 0:27

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Here it means that the derivative is evaluated at (in the first example) $x=c$. Somewhat inconsistently, when you are doing integration, $\left.f(x)\right|_a^b$ will mean $f(b)-f(a)$. –  André Nicolas Feb 5 '13 at 18:04
    
How is that integration? –  Biff Feb 5 '13 at 18:09
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A similar notation is also used for the restriction of a function $f$ to a subset $E$ of its domain: $\left. f \right|_E$. See en.wikipedia.org/wiki/Restriction_(mathematics) –  Ayman Hourieh Feb 5 '13 at 18:18
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@Biff: When you are calculating a definite integral, often you first calculate an indefinite integral (antiderivative), and then do some evaluations. The notation above is used at the final evaluation stage. –  André Nicolas Feb 5 '13 at 18:25
    
@amWhy: Or just use \quad and \qquad... :-) –  Asaf Karagila Feb 6 '13 at 3:13

2 Answers 2

up vote 9 down vote accepted

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. $$F(x)\Big|_a^b = F(b) - F(a)$$.

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It is not integration that is similar: the similarity being the directive to "evaluate _____ at some value x = c". In your case, it's a directive to evaluate the derivative at $x = c$ (or at u = g(c)). In the context of integration, it means evaluate the integral at x = b, subtract the value at x = a. –  amWhy Feb 5 '13 at 18:12
    
Makes sense, thanks. –  Biff Feb 5 '13 at 18:58
    
@valtron I fixed it; thanks! –  amWhy Feb 6 '13 at 13:31

Here, it means "evaluate the derivative at".

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

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