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I was just thinking about how, i.e., if $f\colon\mathbb R\to\mathbb R$ is defined by $f(x) = x^2$, then it's customary to write $$ \frac{df}{dx} = 2x. $$ But since the derivative is itself a function from $\mathbb R$ to $\mathbb R$, shouldn't we write $$ \frac{df}{dx}(x) = 2x? $$ After all, we would typically write $f'(x) = 2x$, not just $f' = 2x$, right?

justin
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    it is just a short notation,you can write both of them – haqnatural Aug 30 '15 at 17:55
  • @VincenzoOliva I wrote $f$ to denote the function, $f(x)$ to denote its value at the point $x$. – justin Aug 30 '15 at 18:00
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    When writing $\frac{df}{dx}$ the $x$ in $dx$ + $d$ instead of $\partial$ tells us that $x$ is the (and only) variable so there is not really a need for the $(x)$. In general if there is no explicit need for such a thing it tends to get omitted. That happens quite often with notation; people tend to go for the minimal way of presenting equations. You are of course free to use it if you feel like it. – Winther Aug 30 '15 at 18:04
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    If $f(x)=x^2,$ then we should write $(df/dx)(a) = 2a.$ But we don't, instead letting $x$ signify two or three different things. We get used to it. – zhw. Aug 30 '15 at 18:10

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For your objection, I prefer: if $u = f(x)=x^2$, then $$ \frac{du}{dx} = x^2 \qquad \text{and}\qquad f'(x)=x^2 $$ Or even $$ \frac{d}{dx}\big(x^2) = 2x $$ Then (if you like) you can avoid both $$ \frac{df}{dx} = x^2\qquad\text{and}\qquad\frac{df}{dx}(x) = 2x $$

GEdgar
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