## What Does dx Mean in Calculus?

The term “**dx**” means *a small change in x.*

More specifically, it’s an infinitesimal (really small!) change in two x-values written in Leibniz notation. We use it in calculus to analyze continuous functions, making the intervals between the x-values smaller and smaller—so small in fact, that the intervals are very close to zero.

Formally, dx is called the differential operator.

## Dx in Derivatives

If you’ve studied limits in calculus, you’ll know that the limit is found by getting *very close* to an x-value.

For example, you might find the limit at x = 1 by looking at what happens when x = .999 or x = .99999. The dx notation describes this limiting procedure and it’s what we use to find derivatives.

You’ll see dx in various forms, including in this notation (d/dx) which means “**take the derivative with respect to x**.”

For example, if you see the following formula:

take the derivative of the function f(x) = 3x – 2.

## What Does dx Mean in Calculus with a ∫?

When you see a ∫ and a dx, it means to integrate. The dx part of an integral tells you *which* variable to integrate; The *x* in *dx* tells you to **integrate with respect to x.** In other words, you’re trying to find the area under a curve by integrating along the

*x*-axis.

Integrals have the basic notation ∫ f(x) dx:

**The function f(x)**is the integrand—the function you’re integrating.**The integral symbol ∫ and the dx**are placeholders that say “integrate everything in between us.”

Some variations include du and dt. Usually these will match the integrand. For example ∫ u^{2} du.

**d**: Integrate with respect to t (time). Like x, this is usually along the x-axis and often appears in physics problems.*t***d**: Integrate with respect to u. The u here is a placeholder used un u-substitution or change of variables, both of which make finding certain types of integrals easier. You can think of the “u” as meaning*u**unknown*.

In multivariable calculus, the “dx” appears in the same position, along with dy and/or dz. When you see a double integral or a triple integral, this notation will tell you in which order you should integrate. For example, this double integral has “dx dy” at the end of the expression:

That’s telling you to integrate first with respect to x, then with respect to y.

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