It's because an integral means you are summing over a lot of very thin rectangles under a curve. The height of the rectangle is f(x) and the width is called $\delta x$ (These two symbols should be read as a single symbol, it doesn't mean $\delta \times x$).

Apr 25, 2023 · The "dx" in this context represents the width of these infinitesimally small rectangles. In a more rigorous mathematical setting, these concepts are formalized using the theory of limits and real analysis. The "dx" is then understood as a limit of a sequence of changes in x, rather than an actual infinitesimally small quantity.

In the setting of measure theory, "dx" is interpreted as a measure; in the context of differential geometry, it is interpreted as a 1-form. But, for the purposes of elementary calculus, the only role of the "dx" is to tell which variable is the variable of integration.

May 2, 2015 · The symbols d/dx and dy/dx represent derivatives in calculus, indicating rates of change with respect to a variable.

Sep 9, 2020 · But then others told me that "dx" is part of what's being integrated, and they started saying that we're led to believe that its just a delimiter in early courses because it'd be impossible for teachers to introduce "differentials," which is what things like dx and du are, so u-substitution isn't just a mnemonic, and the multiplication is ...

Historically, calculus was framed in terms of infinitesimally small numbers. The Leibniz notation dy/dx was originally intended to mean, literally, the division of two infinitesimals. The Leibniz notation $\int f dx$ was meant to indicate a sum of infinitely many rectangles, each with infinitesimal width dx.

Sep 16, 2024 · Is the purpose of the derivative notation $\\frac{d}{dx}$? strictly for symbolic manipulation purposes? I remember being confused when I first saw the notation for derivatives - it looks vaguely like

$$\left(\frac{d}{dx}\otimes\frac{d}{dx}\right)x,$$ and the expression $\frac{d}{dx}\otimes \frac{d}{dx}$ lives in the tensor algebra, rather than in the exterior algebra. The tensor algebra is not anti-commutative, and in fact we do not have $\frac{d}{dx}\otimes \frac{d}{dx} = 0 $ unless $\frac{d}{dx}=0$.

Having a dx alone is almost an abuse of notation, in a way. The concept of “tiny change in y over tiny change in x”, which you can think of as a small slope, might be written as Δy/Δx. In contrast, dy/dx is defined as what you get if you take the limit of that fraction as the change in x becomes infinitesimally tiny.

Apr 8, 2018 · Derivatives: First off, let's start with the derivative. This is commonly referred to as an "instantaneous rate of change", which is a complete oxymoron.