I've noticed that my terminology is a bit haggard. I do math on my own so I'm not entirely sure how everyone else refers to things and so I need a check.

so is this correct: $\lim\limits_{\delta x \to 0}\frac{\delta y}{\delta x} = \frac{dy}{dx}$

Where say, $\delta y$ is the change in distance and $\delta x$ is the change in time and as ${\delta x}$ approaches zero, the whole thing approaches the derivative $\frac{dy}{dx}$.

Would this be the correct notation and, while I'm here, is there a quick reference somewhere online for MathJax notation?

Also, is $\frac{\delta y}{\delta x} \equiv \frac{\Delta y}{\Delta x}$, or does each delta mean something different? Is there a convention here?

  • $\begingroup$ The way most people do things, $dx$ isn't a number, so you're going to have to be more precise about what you mean. $\endgroup$ – Qiaochu Yuan Jan 4 '12 at 1:51
  • $\begingroup$ A d with its top curled to the right isn't really a d but the Greek lowercase letter delta: $\delta$ (\delta). Is that what you mean? $\endgroup$ – user856 Jan 4 '12 at 1:57
  • $\begingroup$ @Rahul, thanks. That's what I was looking for. I've edited my question. Is this the way you would explain the limit of the changes as dx approaches zero? Some math books I've read really try to emphasise this distinction. $\endgroup$ – Korgan Rivera Jan 4 '12 at 2:02
  • $\begingroup$ @KorganRivera: Were you looking for $\delta$ (delta) or for $\partial$ (sometimes called "del", \partial). $\endgroup$ – Arturo Magidin Jan 4 '12 at 4:11
  • $\begingroup$ You'll likely be interested in this (highly regarded) answer. $\endgroup$ – J. M. isn't a mathematician Jan 4 '12 at 9:17

By definition, the derivative of $y=f(x)$ is $y'=f'(x)=\frac{dy}{dx}=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ where $f(x+h)-f(x)$ is the change in $y$ (traditionally denoted $\Delta y$) and $h$ is the change in $x$ (traditionally denoted $\Delta x$). So it's ok to write $\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}$. "Everyone" will know what you mean. For the in's and out's of treating $dy/dx$ like a fraction see this question [Edit: Oops! Wrong link! Fixed].

Now as for $\delta y$ and $\delta x$ (lower-case delta: $\delta$ vs. upper-case delta: $\Delta$)...this usually has a different meaning. Check out: Functional Derivative

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  • $\begingroup$ Thanks Bill, that's what I thought. I've seen $\delta$ and $\Delta$ used interchangeably in some textbooks. But you're saying that the $\delta$ is used when differentiating functions with respect to other functions? $\endgroup$ – Korgan Rivera Jan 4 '12 at 2:32
  • $\begingroup$ As with any notation, different people use it differently. But...usually $\delta$ gets used in variational calculus. If you see it used elsewhere, be careful. It's not standard. $\endgroup$ – Bill Cook Jan 4 '12 at 2:37
  • $\begingroup$ Oh, and more or less "Yes" to your question. :) $\endgroup$ – Bill Cook Jan 4 '12 at 2:38

Warning: This is an editorial! This is a taste issue. I prefer

$$f'(x) = \lim_{h\to 0} {f(x + h) - f(x)\over h} = \lim_{t\to x} {f(t) - f(x)\over t - x}.$$

Either says: the limit of the slopes of the secant lines is the slope of the tangent line. I have never liked the "false fraction" of $dy/dx$. I prefer to think that

$$f(x + h) = f(x) + f'(x)h + o(h).$$

In fact, this last form gives the definition of the derivative that abstracts to many dimensions and to the derivative behaving as linear transformation.

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