# When does $d$ mean a differential and when does it mean a derivative? [duplicate]

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.

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.