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My understanding in univariate case is:

Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$.

Then, $\frac{dy}{dx}|_{x=x_o}=f'(x)|_{x=x_o}$=$\lim_{\Delta x \to 0}\frac{\Delta f}{\Delta x}$, where $\Delta x=(x-x_o)$ and $\Delta f=f(x)-f(x_o)$.

Part of confusion is when I see something like: $dy=f'(x)dx$.

What is happening here exactly? Why some books move around $dy$ and $dx$ as if they are discrete changes? Can somebody explain the difference between discrete changes and differential changes with clarity?

My understanding of $f'(x)$ at $x_o$ is the best affine approximation of $f$ at that point. Then, I use the term "sensitivity", because it is my understanding the derivative (or slope here) provides the information how sensitive function value at that point is to the very little change in $x$.

Then, what does $dy$ mean? Doesn't $dy$ also represent infinitesimal change? So I am confused difference between $dy$ and $\frac{dy}{dx}$ as well.

The terminologies that are used "instantaneous rate of change" or "slope" or "best affine approximation" or "how sensitive f is with respect to the argument at that point" amplifying the confusion.

Please help.

Frank Swanton
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1 Answers1

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The expression $\frac{dy}{dx}$ is on solid footing; it is interpretable as the slope of the function $y(x)$ or the instantaneous sensitivity of $y$ to a tiny change in $x$.

It is convenient to write $dy$ and $dx$ as if they were numeric quantities and manipulate them accordingly. Thus for example, to solve the differential equation $$ \frac{dy}{dx} = y^2 x $$ you manipulate $$ \frac{dy}{y^2} = x \, dx\\ \int\frac{dy}{y^2} =\int x \, dx\\ -\frac{1}{y} = \frac{x^2}{2}+c \\ y = -\frac{1}{\frac{x^2}{2}+c} = \frac{2}{x^2+k} $$ And this works, but what you are really doing in the first step is a stylized shorthand for saying that taking the limits, in a suitable way, of $x$ times the change in $x$ and of $y^{-2}$ times the related change in $y$, those two limits are equal.

This can be made more rigourous in these simple manipulation cases by the field of hyper-reals, where infinitessimals are treated as part of the number system. But hyperreal analysis has gone out of fashion because it does not resolve all issues of this type anyway.

In short, you need to look at equations that use $dy$ and $dx$ as if they were numbers, as shorthand for statements about limits. This can give you peace of mind, while allowing you to take the easy shortcuts this approach enables.

A caveate: Whatch out when partial derivatives appear!!

Mark Fischler
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