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Say we have $f(x) = x^2$. Let $g(x) =\frac d{dx} f(x)$. This, of course, is $2x$. Therefore, we have $g(3) = \frac d{dx} f(3) = 2(3) = 6$. However, if we let $h(x) = \frac d{dx} f(3)$, we have $h(x) = 0$ for all $x$. In particular, we have $h(3) = \frac d{dx} f(3) = 0$. There is clearly something ambiguous about $\frac d{dx} f(3)$. On one hand, we mean the $y$ value we get then we plug in $3$ into the derivative of $f(x)$. On the other hand, we mean the derivative of $f(3)$, a constant. Is there a way to make this distinction more clear? I have not explored every calculus notation, but for something like prime notation, we might write $f'(3)$ for the former case and $f(3)$ for the latter. How about for Leibniz?

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    $\begingroup$ $$\frac{d}{dx}\big|_{x = 3}$$ $\endgroup$ – user296602 Oct 3 '17 at 2:31
  • $\begingroup$ You can do other whimsical things with this notation. In the 19th century one can see $$\frac{d^n 0^m}{d0^n}$$ used for $$\frac{d^n x^m}{dx^n}\Big|_{x=0}$$ $\endgroup$ – GEdgar Oct 3 '17 at 13:47
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This is precisely why we don't write $\frac{d}{dx}f(3)$. Instead, you are more likely to see $f'(3)$ or $\frac{df}{dx}\Bigr|_{x=3}$.

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  • $\begingroup$ Or $\frac{\mathrm{d}f}{\mathrm{d}x}(3)$. $\endgroup$ – Xander Henderson Oct 3 '17 at 2:34
  • $\begingroup$ @XanderHenderson I personally don't see that notation very often. $\endgroup$ – Simply Beautiful Art Oct 3 '17 at 13:16
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    $\begingroup$ @SimplyBeautifulArt I see it with some frequency, usually in the context of Radon-Nikodym derivatives. It is not the most common notation, but it is relatively clear and unambiguous, and it does show up. $\endgroup$ – Xander Henderson Oct 3 '17 at 13:34

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