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In a lot of books that I've read about calculus the authors will sometimes use the operator $d$ to mean a differential (an "infinitesimal" change), and sometimes they'll use it to mean an actual derivative. For example they'll write: $d(x_1x_2)=dx_1x_2+ x_1dx_2$ or they'll write $d x^2 = 2x$ when if they at least write it as $d x^2 = 2xdx$ it would make more sense as a derivative notation.

When should one interpret this $d$ operator as a "small change" or conversely as a derivative? This mix-up gets even more confusing when manipulating differential equations.

Thanks.

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The notation $dx^2=2x$ is wrong and might confuses you.

The abstract derivative as a limit of differential quotient comes first. The following notation is all the same: $$ \lim \limits_{x\to a} \frac{f(x)-f(a)}{x-a}\equiv f'(a)\equiv \left.\frac{dy}{dx}\right\vert _{a}.$$ However, since a derivative is computed by the quotient, we might design a new notation of differential that looks like a fraction but the action when the numerator and the denominator come together is differentiation.

The new notation $dy$ is defined to be a function of the point $a$ and a small increment $\Delta x$, $$dy:=f'(a)\cdot \Delta x.$$ You can use $dx$ instead of $\Delta x$ if you wish, though. From this definition and linear approximation $f(a+\Delta x)\approx f(a)+f'(a)\cdot \Delta x$, we have $$dy\approx f(a+\Delta x)- f(a)$$

You see that $dy$ is now viewed as a change in the value of $y$ (also $f$) at the point $a$ after an increment of $\Delta$ as how we designed it should be.

To sum up, the differentials $dy$ or $dx$ is to be viewed as a small change while $\frac{dy}{dx}$ is to be viewed as a derivative.

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