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Is it mathematically correct to write $$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx},$$ rather than $$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}?$$ If not, what is the difference? If so, why isn't this notation used from the beginning? My feeling for the latter is that it would align the derivative more with the inverse of the indefinite integral.

John Molokach
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    It might look odd, but actually you can also write $$f'(x) = \lim_{John \to 0} \frac{f(x + John) - f(x)}{John},$$ as long as you are aware that John is just a nonzero number which is taken to be arbitrarily small, just like $h$ itself. –  Oct 21 '15 at 12:58
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    Depends on what you are interpreting $dx$ as. If it is just a name like John Ma said is fine. If it is a form then this doesn't make sense. – Aaron Maroja Oct 21 '15 at 13:04
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    why is it taught that way? might be a question for http://matheducators.stackexchange.com/ – null Oct 21 '15 at 13:06
  • @null, very good point - I hadn't even thought of seeing if there was a stackexchange for that! I'll post there. Thanks! – John Molokach Oct 21 '15 at 13:14

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We do have the notation $f'(x) = \frac{df}{dx}$, which as you say, "aligns the derivative more with the inverse of the integral". However, in its usual usage, the particle $dx$ is not in itself a number (see this question for more on that), so using $dx$ like that in a limit is misleading.

A better usage might be as follows: $$ \frac{df}{dx} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x} $$ We could also apply this to integrals: $$ \int_a^b f(x)\,dx = \lim_{\Delta x \to 0} \sum_{k=1}^{N(\Delta x)} f(x^*_k)\,\Delta x$$

Community
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Ben Grossmann
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    In some contexts, it is common to use $\Delta x$ rather than $h$. However, some students are thrown off by the fact that $\Delta x$ consists of two characters. – Ben Grossmann Oct 21 '15 at 13:14
  • thanks for the clarification, and also for introducing me to Robinson's "Nonstandard Analysis." I rather like the prospect of using this book: http://www.math.wisc.edu/~keisler/calc.html – John Molokach Oct 21 '15 at 14:01
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No it doesn't matter, as pointed out what you write doesn't matter, as long as the concept gets across, $dx$, $f$, $john$ $\oplus$, you can use any of them, $h$ is just the common one and used because everyone recognises it but in certain context, to make clear, you must use other letters as you're doing more than one thing at once.

Zelos Malum
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It would be more common to use $\delta x$, to make it clear that it's "a small quantity" rather than the infinitesimal $dx$. Do note that you can't necessarily just manipulate $dx$-the-infinitesimal in the obvious ways: for instance, $$\dfrac{\partial a}{\partial b}_c \dfrac{\partial b}{\partial c}_a \dfrac{\partial c}{\partial a}_b = -1$$ rather than the $1$ you might expect, so funky things can happen with infinitesimals.

Patrick Stevens
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