But what is “dx” really? Calculus terms explained

academics calculus math
By Jane

The symbol “dx” comes up everywhere in calculus. For example:

  • If y is a function of x, then we sometimes write the derivative of y with respect to x as the following:

  • When we write indefinite integrals, they are written as:


  • When we write definite integrals, they are written as:

But what is “dx” really? It’s more than just notation! In this post, we’ll explore the meaning of “dx” and try to get a better understanding of some of the symbols that we often see in calculus.

Calculus as the study of infinitesimal change

Calculus is the study of continuous change, or infinitesimal change. To get an idea of what this means, let’s consider the following: suppose that you’re running a race. You begin running at time x=0 seconds and then track your displacement as a function f(x). Your displacement function f(x) is then a continuous function that varies over time.

Suppose now that you wanted to know your speed at time x=10 seconds. That is a calculus question since you are looking for a rate of change at one particular time. How could you do this? Well, you might estimate your speed at time t=10 as our average speed between time x=10 and x=11, which can be expressed as:

But you could get a better estimate by choosing a smaller time interval, say from x=10 to x=10.1 instead, or even better from x=10 to x=10.01.

Really, what you want to do is to take the limit as the size of your time interval goes to zero. That is, you want to take an infinitesimally small change in x. Then, by the definition of a derivative, your speed at x=10 seconds is:

Screen Shot 2020-02-05 at 3.24.36 PM

We should think of h going to zero as taking smaller and smaller increases in x when we take the average speed from time x=10 to time x=10+h.

This is where “dx” comes in.

"dx" is an infinitesimal change in x. We can think of "dx" (read as dee-ex) as an infinitesimally small change in x. The "d" in "dx" should remind you of a delta ∆, which is the symbol for change. "dx has no numerical value. Rather, it captures this idea that occurs a lot in calculus of taking the limit of smaller and smaller interval sizes to figure out something precisely about a continuous function.

We can see this if we revisit our speed example from earlier. When computing our derivative

Screen Shot 2020-02-05 at 3.26.02 PM

The bottom of this fraction is (10+h)-10 as h goes to zero, which is an infinitesimally small change in x. We might therefore think of the denominator as h goes to zero as dx. If we let y=f(x), then the numerator of this fraction is f(10+h)-f(10) as h goes to zero, which is an infinitesimally small change in y, or dy. Putting this all together, we recover the notation:

Screen Shot 2020-02-05 at 3.26.41 PM

That is, the derivative of f(x) is the quotient of an infinitesimal change in y over an infinitesimal change in x. Put more precisely, it is exactly the limit of the change in y over the change in x over smaller and smaller changes in x. The “dx” and “dy” notation just captures this limiting procedure and expresses it as an infinitesimal change in x or y instead.

“dx” as seen in integrals

Another place where "dx" is often seen is in integrals. Let's focus on definite integrals. What does "dx" mean in a definite integral?

"dx" here is still an infinitesimal change in x. To see why it's there, we should think of the integral as a signed area and as the limit of Riemann sums. We recall that to compute a left Riemann sum of f(x) from x=a to x=b with n intervals, we let the following be true:

Then we take:

...where x takes values a, a + ∆x, a + 2∆x,..., a + (n - 1)∆x = b - ∆x.

Then, as we let n go to infinity, ∆x gets smaller and smaller, and the Riemann sum converges in value to the integral, which is the signed area under the curve f(x) between x=a and x=b. The below picture (from the Wikipedia article on Riemann sums) shows this convergence process:

Now, we can see where the notation for the integral comes from. The integral sign ∫ is the continuous version of the sum sign ∑. The bounds of integration from a to b are like the first and last x values for the summation. And the dx is the infinitesimal version of ∆x, what we get when we take smaller and smaller step sizes in x.

Another way to think about this is that in our integral, we are summing up infinitesimally thin rectangles with height f(x) and width dx to compute our signed area precisely.

Hopefully, the symbol “dx” is a little less mysterious now!


academics study skills MCAT medical school admissions SAT expository writing college admissions English MD/PhD admissions GRE GMAT LSAT chemistry writing strategy math physics ACT biology language learning test anxiety graduate admissions law school admissions MBA admissions interview prep homework help creative writing AP exams MD study schedules summer activities history personal statements academic advice career advice premed philosophy secondary applications Common Application computer science organic chemistry ESL PSAT economics grammar test prep admissions coaching law statistics & probability supplements psychology SSAT covid-19 legal studies 1L CARS logic games reading comprehension Spanish USMLE calculus dental admissions parents research Latin engineering verbal reasoning DAT excel mathematics political science French Linguistics Tutoring Approaches chinese DO MBA coursework Social Advocacy academic integrity case coaching classics diversity statement genetics geometry kinematics medical school skills IB exams ISEE MD/PhD programs PhD admissions algebra astrophysics athletics biochemistry business business skills careers data science letters of recommendation mental health mentorship quantitative reasoning social sciences software engineering trigonometry work and activities 2L 3L Academic Interest Anki EMT English literature FlexMed Fourier Series Greek Italian Pythagorean Theorem STEM Sentence Correction Zoom algorithms amino acids analysis essay architecture argumentative writing art history artificial intelligence cantonese capacitors capital markets cell biology central limit theorem chemical engineering chromatography climate change clinical experience cold emails community service constitutional law curriculum dental school distance learning enrichment european history finance first generation student fun facts functions gap year harmonics health policy history of medicine history of science information sessions institutional actions integrated reasoning intern international students internships investing investment banking logic mandarin chinese mba meiosis mitosis music music theory neurology operating systems phrase structure rules plagiarism poetry pre-dental presentations proofs pseudocode school selection simple linear regression sociology software study abroad teaching tech industry transfer typology units virtual interviews writing circles