The position, as you queried, can be realized by those who are missing solid, basic arithmetic and algebraic skills. The foundation for every high level math class depends upon solid basic arithmetic and algebra skills. When those are missing, it is difficult to add on the layer of calculus. However, once those basic skills are developed (no choice to do so in order to be successful in calculus), one is more successful and confident in calculus.

The position, as you queried, can be realized by those who are missing solid, basic arithmetic and algebraic skills. The foundation for every high level math class depends upon solid basic arithmetic and algebra skills. When those are missing, it is difficult to add on the layer of calculus. However, once those basic skills are developed (no choice to do so in order to be successful in calculus), one is more successful and confident in calculus.

I believe it is so because conceptually it is difficult to visualize the infinitesimal.

In my opinion, the main difficulty of teaching/learning calculus that the students do not hold the basics of the previous learning as some students find difficulty in simplifying and factorizing the algebraic quantities, trigonometric functions, operations on fractions, ... etc. In short, the main problem lies in the prerequisite.

The content of a typical Calculus course consists of two quite different things: a lot of applications and a few ideas. Historically, the applications came first and mathematicians understood the difficulties involved in attacking them individually. Only then did Newton and Leibnitz develop the insights which made common formulations possible and common strategies possible. We very seldom make this distinction explicitly in the classroom so the students find the ideas difficult because "too theoretical";

rather than a way to reduce computation and simplify modeling. The applications are "difficult" because too repetitive, with the emphasis on being expected to work proficiently with techniques which they saw originally without any suggested that they would be helpful. Perhaps we need to order the material to emphasize how tools get used, so a tool is presented when it can be motivated (when applications can be seen as valuable with a difficulty which the tool can handle) rather than historical order or didactic order (when the relevant definitions can be presented).

Dear Asia Majeed,

I do not agree with your claim. :)

If calculus is the hardest part, then what is the simplest part?

If calculus is hard, then mathematics, in general, is very hard.

On the contrary, Calculus is the simplest part comparing with other courses. Using calculus of limits, differentiation, and integration to find areas, lengths, tangent lines, and their applications, etc., by applying direct formulas. It is interesting, where the student doesn't need the deep concepts of mathematics as other elementary university courses ( complex analysis, differential equations, numerical analysis,..)

Best regards

Calculus has traditionally been the first or one of the very first mathematics courses most undergraduates take in the first year of studies. For many students it turns to be the only mathematics course towards bachelor degree. Since transition from school mathematics to university mathematics has always been difficult, students with insufficient preparation find standard Calculus course hard, whereas those with a good command of school mathematics do not find it particularly challenging compared to other mathematics courses. Students who do not take other mathematics courses may lack motivation to study Calculus and may set "only passing" the course as the main goal; this attitude quite often plays bad jokes on such students...

I had intended to add a comment like that of Prof Rogochenko, but he wrote it first. In any reasonable absolute sense, Algebraic Topology is much more difficult than Calculus, but students taking that find it less difficult because they are already much more sophisticated and know what to do.

Besides those written above by our colleagues, I would like to add the fact that most of the students do not know computing derivatives of elementary functions. Some of them do not understand the notion of "differential of a function". When passing to several variables, the situation becomes more complicated. Now if we refer to Integral Calculus, they should also have some knowledge of analytic (and differential) geometry, in order to understand multiple integrals, integrals on curves and surfaces, integral formulas. Usually, most of the students have not such a background. The connection between equations, formulas and drawings is a difficult task for them. However, about 10% of my students do have enough knowledge and are able to understand new information on these subjects.

I suspect that "calculus is considered the hardest part of mathematics" only by those who do not study maths beyond calculus.

If you understand the fundamental concept of limit of a function, then Calculus could be easily learned.

The essence of calculus is its use of function concept and limiting processes(limit, infimum, suprema, etc). But the notion of limit is not taught at elementary school level, unlike algebra and geometry, which are taught beginning from KG level. As the foundations of calculus (limit, function and related concepts) are themselves dependent on several concepts from algebra and geometry, it is difficult/impossible to start calculus from early education/elementary school level. This could be one reason for the hardness of calculus relative to other branches of mathematics.

The topic i.e. 'Why calculus is considered as a hardest part of mathematics especially by the university students? It is a most common problem which has been seen among university students'. As a civil engineer, I find this to be a very interesting subject: We (that is all disciplines studying engineering or survey) had to do 1st year Mathematics at University (in the 1st year of study of a 4 year degree) as well as 2nd year Mathematics thereafter. First (1st) year Mathematics included a revision of ALL school mathematics in the first two (2) weeks and Calculus commenced with a thorough and detailed use of the functional concept and limits (as so termed by Tadesse Bekeshie Gerbaba). In South Africa, both the use of 'function concept and notion of limit, or theory of limits, including infinity, is introduced and used  in the last two (2) years of high school i.e. those two (2) years before commencing 1st year University. Despite the modern trend of 'attempting to steer away' from Geometry, the content of Mathematics as a subject in South Africa still includes Algebra, Arithmetic, GEOMETRY, Trigonometry,  and Statistics AND in the last year of high school their is a fair section on Calculus (right up to basic integration etc.).

This being so, the Student is more than adequately provided with the 'tools' to tackle both Calculus and Pure Mathematics (even when Calculus is a component of that) in the 1st year at University. The reason why it could be deemed to be SO HARD is that, in my opinion, the rate of knowledge TRANSFER in the last year of school is much slower than that at the first year of University: In fact many orders of magnitude slower. At University, a section in Calculus (or Mathematics) is covered only once i.e. the lecture is taken 'as read', tutorials are given, questions are asked in tests and then in an examination. It all depends on the Student, how many subjects he/she is taking etc. In my case, the only way to master Calculus (and Mathematics) at University was NOT to proceed to the next section until I had fully understood and mastered the previous section AND with a lot of hard work as well. That is the key because Calculus itself is NOT hard if a person PROGRESSES only when they have made themselves fully acquainted with previous sections. The same applies to Complex Numbers and Series which (depending on the Mathematics Syllabus) also forms a part of Calculus as one progresses to more advanced aspects of Complex Numbers and Series.                   

Concur with Alan's points about the rate at which material is taught in universities, as opposed to school, and I agree with those who pointed out that calculus is considered "most difficult" only by those who never go beyond calculus, in their math studies. Which is the majority.

But there is another point too. The concepts of calculus are not so difficult to grasp, but the mechanics very quickly become really difficult, at least for most for most people. You quickly run out of tricks, as expressions get more complicated. Perhaps students would be less anxious if they were told that in practice, problems they would be required to solve are a small set of solvable problems. Not to let their imaginations run wild, causing panic, say, before a closed book exam.

Mathematics is a discipline of abstraction and by virtue of being abstract it is not easily visible or understandable by many students. Mathematics is a subject which is becoming more closer to the one who gets closer to it, who develops love affair with it and continues doing it. Mathematics is like the human shadow beneath a light bulb hanging on a pole, the closer you get to the light pole, the shorter and closer to the light pole your shadow is and the farther away you go, the longer, unreachable and clearly unidentified your shadow is.

Calculus or any part of mathematics from the learning end is exactly the same, it will not be difficult to those who study it up close unlike to those who want to know it from afar. Mathematics is not a spectator game to enjoy with watching but a game of practitioners to develop love affair with. In fact it is a logically structured correctly piloted field of study, which requires a love affair with its abstractness, structures of consistency and immense powers to solve problems.

On thinking it over, I don't agree with the topic: that "calculus is considered as a hardest part of mathematics especially by the university students."

If one evaluates difficulty objectively by success rate (passing the course with a C or better), then the "difficulty" is correlated most strongly with a mismatch between the course expectations and student preparation -- in universities peaking in remedial courses (either general distribution requirements or "Statistics for Psych majors or...).

I do not agree about "calculus is considered as a hardest part of mathematics especially by the university students."

The statements needs to be corrected as "calculus is the main part of mathematics requested to the university students." Algebraic homology, nonlinear logics are not simple but they are not usually requested to physics, engineering students.

Most applications need calculus, as differential equations, geometry...

And there is a level in which the arguments become intertwined... number theory, cryptology, category theory, differential geometry, functional analysis...

From my experience, the derivative should always be introduced related to the physical concept of speed, and accompanied by examples from physics. For example, in the differential equation of the level dynamics in a water tank with cylindrical geometry, which is filled with water and can be also emptied: it is easier to understand the fundamental concept of variation, provided that the concept of limit is firstly understood. Also, the concept of input and output of a dynamical system, modelled by a differential equation, can be understood by explaining the causality effect: the difference between input and output flows is balanced by the water level variation in time, so the water level, that is the variable of the differential equation, is always the output of the dynamical system, etc.

To move from linear (y=mx+b) concepts in 2-D to nonlinear ones in 3-D (including rational equations) numerically, graphically, and conceptually is a new way for students to "see" and think about mathematics. Applied math (as I see calculus) calls for a change in mindset that incorporates and successfully uses algebra and geometry. When most students take calculus (at least where I teach) it is usually one or two years after they took geometry. All of a sudden, they are expected to recall and apply formulas for areas (including surface area) and volumes of various shapes (cylinders, cones, rectangles (boxes with/without lids), spheres, etc.) and combine them with new terminology from calculus such as rates of change (while holding one variable constant). Then, one adds parametric and vectors to the calculus equations, students are overwhelmed because their lack of understanding or developing the basic skills from algebra and trigonometry prevent them from applying those skills at a higher level. They blame "calculus" for being the most difficult when in fact it stems from a gap in prior concepts.

I suspect the "hardest course" is inevitably the highest level course taken --- which, for most students in high school or college is typically Calculus. For students who then take Linear Algebra or Differential Equations (or PDE) the evaluation is then superseded by that. Etc.

That is spot on Natalie Carter Johnson. In essence, it is the gap between high school mathematics which (in South Africa) is arithmetic, algebra, geometry, statistics and introduction to calculus all of which are examined in various ways in two (2) three hour papers in the National Senior Certificate (or Matriculation) examination. You will be surprised - basic calculus (including limits, differentiation etc.) is always asked as the last question and generally allocated a very, very generous 30 marks out of 120 marks. This happens ALL the time: This being so, and introduction to calculus being the LAST SECTION to be taught at high school near the very end of the 2nd Semester, the students (scholars) merely write it off - just too much trouble to get to grips with something easy. This happened to both my daughters - I pointed out to them to get to grips with the 'very easy introduction to calculus' AND do that question first. It would mean 30 marks out of 120 marks very quickly at the beginning of an examination. Bottom line - it is all a very simple mindset that has to be bridged. Best regards. Alan Chemaly (a Specialist Engineer: Dams, who still loves his school mathematics and regularly follows the development of teaching thereof).       

Analysis/calculus seems to be hard, since presented without real applications. And they are in natural sciences. But then one needs teachers with more wide knowledge, than just math. Simplest solution would be to prepare teachers of both math and some mixture of phys/chem/bio/econ/...(some domainswhere math is applied). This is an utopia, unless deep changes of understanding the need of mathematical modeling would become real among our leaders. But what can we do with those politicians who are proud of being weak in math, since this feature didn't disturb their carrier?

I support 100 percent Joachim's contribution. Modeling should be an ultimate aim of calculus. Without applications calculus will fade away.

We should distinguish between Analysis and Calculus. Analysis is a rigorous discipline of continuous mathematics which is a backbone of calculus, that is, esp. in the USA, a standard series of three 4-credit courses taught for STEM majors. Usually, the emphasis on the proofs is not so strong, but a proper teaching with applications to modeling is done only by the best professors.

My professional experience indicates that both professors, Joachim Domsta and Thomas I. Seidman belong to this group.

In my (humble) opinion, another problem is that a full motivation and explanation of the importance of calculus in science would need another course of mathematics... and for persons with a mathematical background.

In a course of calculus, some teachers may introduce the first lecture with a bit of history of maths (Newton, Maxwell and his equations, hints about financial maths) and some anecdotes; but this has to be done only in the begin. Well: and the students need to verify this part, too.

And eventually a teacher can rightly suppose that students have already some ideas on the courses they choose, and they have to know that physics, engineering without mathematics do not work. Finally, justifying a highly and traditionally necessary course is sometimes _really_ boring. (And you end up to thinking that the interlocutor is highly ignorant!)

I think, calculus/analysis has not new teaching methods like abstract algebra and so on.