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I've been stuck on a few calculus concepts the last day or so. One of which is how we apply calculus to physical phenomena and the assumptions that go into it. Say I have a derivative $\frac{dy_1}{dx_1}$. I want to derive a separate derivative $\frac{dy_2}{dx_2}$. $y_1$, $y_2$, $x_1$, $x_2$ are all physical phenomena and thus we can observe some things about them, such as $y_1$ = $2\times y_2$ usually or something like this. Within the context of these relationships, is it valid to then say that

$$\frac{dy_1}{dx_1} \times \frac{dx_1}{dx_2} \times \frac{dy_2}{dy_1} = \frac{dy_2}{dx_2}$$

This feels bad since derivatives are not exactly constants that can be canceled out like this and thus this kind of operation doesn't seem generalizable. But I think it might work in my example? I think I am missing a fundamental understanding of derivation and could use some guidance.

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Going to provide a couple details to see if any of these are helpful.

The first one, I'd recommend you went and read https://maa.org/press/maa-reviews/calculus-an-intuitive-and-physical-approach , chapters 1 through 7 cover the content of your question I believe.

Now onto your question. I'm going to change the language a bit. It seems like you are thinking of observing a system and $y_1, x_1, y_2, x_2$ are some set of observable quantities (and there could be more). You perform some measurements, or have some hypothesis, and the expression $$ y_1 = 2y_2 $$ comes up. If you wanted to calculate the derivative of $y_2$ with respect to $x_1$, then the chain rule is what you need. We already have your expression $y_1 = 2y_2$ or $y_2 = \frac{1}{2}y_1$. Let's say that we also knew that $y_1$ was a function of $x_1$, or $y_1 = y_1 (x_1)$. Then the derivative would be as follows $$ \frac{d y_2}{d x_1} = \frac{d y_2}{d y_1} \frac{d y_1}{d x_1} $$

Notice that I changed your question a bit, we are taking the derivative of $y_2$ with respect to $x_1$ to show how the interdependence comes up when you take a derivative (and this is explained by the chain rule). Usually, given the notation that you are using, one could assume that $y_2$ is more simply related to $x_2$ so that taking a derivate would be easier, and possibly not need the chain rule.

For modeling of more realistic systems, there may be even more complicated relationships so that, for example, $y_2 = y_2 (y_1, x_1, x_2)$. But that goes into multivariable calculus (linear algebra would also come in handy). In such cases you are looking at partial derivatives but the chain rule will still more or less look the same.


Also forgot to mention: I'd shy away from calling this sort of thing a change in the units of the derivative because a change in units is an exercise that is done in a manner that doesn't have to do with the chain rule. Another way of saying it: changing units of the thing you are observing is based on equvalences that you know or measure, but those don't have to do with derivates. For example, there is no need for the chain rule in going from metres to inches.

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