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?
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$\begingroup$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
$\endgroup$ 3 $\begingroup$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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