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When I first learned calculus, I was taught that $'$ for derivatives was only a valid notation when used with function notation: $f'(x)$ or $g'(x)$, or when used with the coordinate variable $y$, as in $y'$.

But I have seen on a number of occasions, both here and in the classroom, where it will be used with an expression. E.g. $(x+\frac{1}{x})'$ to mean $\frac{d}{dx}(x+\frac{1}{x})$. It has always been my understanding that this notation is not considered valid because it doesn't indicate what the independent variable that the expression is being differentiated with respect to is. E.g. in $(ax+bx^2)'$, the variable could be $a$, $b$, or $x$. This problem also exists with $y'$ but I figured this was an exception because $y$ and $x$ usually represent the coordinate axes so it can be assumed that the independent variable for $x$ is $y$ when taking $y'$.

So is this notation valid, just putting a $'$ at the end of an expression?

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    $\begingroup$ If it is clear from context, it is OK. One might just write $(ax+bx^2)_x^\prime$ instead, say. $\endgroup$
    – Pedro
    Jun 17, 2013 at 23:56

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What you're seeing is a "shorthand" an instructor or such may use in the process of computing the derivative of a function with respect to $x$. Usually when you seem something like $(ax + bx^2)'$, it's assumed from the context that we are taking the derivative of the expression, with respect to $x$. That is, "$(ax + bx^2)'$" is taken to mean "evaluate $\,\frac d{dx}(ax + bx^2)$", just as one assumes from context that $y'$ refers to the derivative, $f'(x)$, of $y = f(x)$.

I prefer to stick with $f'(x)...$ or $y'$, using $\frac d{dx}(\text{some function of x})$ when evaluating the derivative of a function with respect to $x$, particularly when trying to convey information to another person. (On scratch paper, or in my own work, I might get a little informal and slip into using a "prime" to abbreviate what I'm doing.) But I would prefer the more formal or "official" conventions/notations were used in "instructive contexts", to avoid confusion or possible ambiguity.

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