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What Does dx Mean in Calculus?

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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.”
what does dx mean in calculus?

For example, if you see the following formula:
what does dx mean in derivatives
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 ∫ u2 du.

  • dt: Integrate with respect to t (time). Like x, this is usually along the x-axis and often appears in physics problems.
  • du: 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 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:
what does dx mean in calculus

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

Stephanie Glen. "What Does dx Mean in Calculus?" From Calculus for the rest of us!

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